Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 55-88.

Decomposition of Rings under the Circle Operation

Clare Coleman David Easdown

School of Mathematics and Statistics, University of Sydney NSW 2006, Australia e-mail: cec@maths.usyd.edu.au de@maths.usyd.edu.au

Abstract. We consider rings S, not necessarily with 1, and develop a decomposition theory for submonoids and subgroups of (S, ∘) where the circle operation ∘ is defined by x ∘ y = x + y − xy. Decompositions are expressed in terms of internal semidirect, reverse semidirect and general products, which may be realised externally in terms of naturally occurring representations and antirepresentations. The theory is applied to matrix rings over S when S is radical, obtaining group presentations in terms of (S, +) and (S, ∘). Further details are worked out in special cases when S = pℤ_{p^{t}} for p prime and t ≥ 3.

1. Introduction and preliminaries

Groups of units of rings with identity are well studied. However many rings arise naturally without an identity. For example, nontrivial rings which coincide with their Jacobson radical never have an identity. Nevertheless, all rings possess groups of quasi-units, that is, elements which are invertible with respect to the circle operation ∘ defined by

x ∘ y = x + y − xy .

Consider a ring S, not necessarily with 1, with multiplication denoted by · or juxtaposition. We refer to (S, ∘) as the circle monoid of S. Denote by S^{1} the result of adjoining 1 to S, which may be done in different ways depending on the characteristic (see, for example, [10, Theorem 2.26]). Then the mapping

: (S, ∘) → (S^{1}, ·) , x ↦ x̂ = 1 − x (x ∈ S)

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

is a monoid embedding, which is an isomorphism when S = S^{1}. An element x ∈ S is called quasi-invertible if there is an element y such that

x ∘ y = y ∘ x = 0 ,

in which case we call y the quasi-inverse of x and write

x′ = y and \overline{x} = 1 − x′ ,

so that, in S^{1},

\overline{x} x̂ = x̂ \overline{x} = 1 .

Put

G(S) = { x ∈ S | x is quasi-invertible } ,

called the group of quasi-units or the circle group of S. When S = S^{1}, denote by G(S) the group of units of (S, ·), in which case : G(S) → G(S) is a group isomorphism.

The Jacobson radical of S, denoted by J (S), may be defined to be the largest ideal of S consisting of quasi-invertible elements. It is easy to see that any ideal of S contained in J (S) forms a normal subgroup of (G(S), ∘). The existence of complements of J (S) and the nilradical in G(S) appears to be a delicate issue, investigated in [7].

Call S radical if S = J (S). The circle group of a radical ring has also been called the adjoint group [40]. Chick [3], [4] investigates, also with Gardner [5], interesting examples of commutative radical rings S in which (S, ∘) and (S, +) are isomorphic. The question of when an abstract group arises as the circle group of a ring, and the interplay between finite generation, nilpotency of the ring and nilpotency of its circle group have been investigated by a number of authors including Ault, Watters, Kruse, Tahara, Hosomi and Sandling [1], [40], [12], [13], [39], [37]. Membership criteria for the circle groups of band graded rings have been investigated by Kelarev [11].

It should be remarked that many authors use as circle operation ∘^{+} defined by x ∘^{+} y = x+y+xy. This does not matter in our context, however, because negation is an isomorphism between the monoids (S, ∘) and (S, ∘^{+}). Both ∘ = ∘^{(−1)} and ∘^{+} = ∘^{(1)} are special cases of the derived associative operation ∘^{(k)}, where k is an integer, defined by

x ∘^{(k)} y = x + y + kxy .

Derived associative operations are characterized by McConnell and Stokes[21]. If k is invertible modulo the characteristic of S with inverse reperesented by ℓ then it is easy to see that (S, ∘) ≅ (S, ∘^{(k)}) under the map x ↦ ℓx for x ∈ S.

In this paper we develop a general decomposition theory (Section 5) for submonoids and subgroups of rings under ∘, in terms of semidirect, reverse semidirect and general products, defined later in this section. Details of the mappings involved in the case of semidirect and reverse semidirect products can best be understood in terms of naturally occurring representations and antirepresentations (Section 4). This theory is applied to obtain decompositions of the circle group of the ring of matrices with entries from a radical ring S (Section 6), yielding a group presentation (Section 7) in terms of (S, +) and (S, ∘), further details of which are worked out (Section 8) when S = pℤ_{p^{t}} for p prime and t ≥ 3.

We establish here some notational conventions used throughout the paper. If M is a monoid then its identity element is denoted by 1 or 1_{M}, and the dual of M is the monoid M^{∗} = {x^{∗} | x ∈ M} with multiplication

x^{∗}y^{∗} = (yx)^{∗} (x, y ∈ M) .

The cyclic group of order n is denoted by C_{n}, written multiplicatively. If G is a group and x, y ∈ G then we write

x^{y} = y^{−1}xy and [x, y] = x^{−1}y^{−1}xy ,

and if H is a subgroup of G then we write H ≤ G. The use of angular brackets varies slightly according to context. If X is a subset of a monoid or group then ⟨X⟩ denotes the submonoid or subgroup, respectively, generated by X. The difference in meaning never causes confusion here. If X is a subset of a ring S then ⟨X⟩_{+} denotes the additive subgroup generated by X, and if X ⊆ G(S) then ⟨X⟩_{∘} denotes the subgroup generated by X under ∘. If Σ is an alphabet and R a collection of relations then ⟨Σ | R⟩ denotes a group presentation. Manipulations of group presentations in the final sections use Tietze transformations, a good reference for which is [23]. In some examples, monoid presentations appear (which are not groups), for which we adopt the notation ⟨Σ | R⟩_{monoid}.

Let S be a ring, x ∈ G(S) and k ∈ ℤ. Denote the kth power of x in (S, ∘) by x^{∘ k}, and note that, since is a monoid homomorphism, (1 − x)^{k} = 1 − x^{∘k}. It is well-known (see, for example [22, Theorem XVI.9]), for p prime and n ≥ 1, that the group of units of ℤ_{p^{n}} is isomorphic to C_{p−1} × C_{p^{n−1}} , if p is odd, or p = 2 and n ≤ 2, and C_{2} × C_{2^{n−2}}, if p = 2 and n > 2. It is easy to see that

(pℤ_{p^{n}}, ∘) = \begin{cases}⟨p⟩_{∘} & if p is odd, or p = 2 and n ≤ 2 ; \\ ⟨2, 4⟩_{∘} & if p = 2 and n > 2 . \end{cases}

If n ≥ 1 then we denote by M_{n}(S) the ring of n × n matrices with entries from S. Note that J (M_{n}(S)) = M_{n}(J (S)). If S is radical then so also is M_{n}(S), whence M_{n}(S) = G(M_{n}(S)) is a group under ∘.

Our development begins by recalling a well-known construction. Let M and N be monoids. Given a monoid antihomomorphism φ : M ⟶ End (N) then we may form the (external) semidirect product

N ⋊_{φ} M = { (n, m) | n ∈ N , m ∈ M }

with multiplication

(n_{1}, m_{1})(n_{2}, m_{2}) = (n_{1}[n_{2}(m_{1}φ)], m_{1}m_{2})

which is easily seen to be a monoid with identity (1, 1) . Dually, given a monoid homomorphism φ : M ⟶ End (N) then we may form the (external) reverse semidirect product

M ⋉_{φ} N = { (m, n) | m ∈ M , n ∈ N }

with multiplication

(m_{1}, n_{1})(m_{2}, n_{2}) = (m_{1}m_{2}, [n_{1}(m_{2}φ)]n_{2}) ,

which is a monoid, and one may verify that

(M ⋉_{φ} N)^{∗} ≅ N^{∗} ⋊_{φ^{∗}} M^{∗} (1.1)

under the map (m, n)^{∗} ↦ (n^{∗}, m^{∗}) for m ∈ M, n ∈ N, where φ^{∗} : M^{∗} ⟶ End (N^{∗}) is the antihomomorphism

m^{∗}φ^{∗} : n^{∗} ↦ (n(mφ))^{∗} (m ∈ M , n ∈ N) .

In both cases above one can easily verify that if M is a group then φ : M ⟶ Aut (N) . If M and N are both groups and φ : M ⟶ Aut (N) is an antihomomorphism then one verifies that N ⋊_{φ} M is a group (see also (1.5) below) and

N ⋊_{φ} M ≅ M ⋉_{ψ} N (1.2)

under the map (n, m) ↦ (m^{−1}, n^{−1})^{−1} for m ∈ M, n ∈ N, where ψ : M ⟶ Aut (N) is the homomorphism defined by mψ = m^{−1}ϕ for m ∈ M. This accords with (and can be deduced from) isomorphism (1.1) because every group is isomorphic to its dual under the inversion mapping.

For the development of the theory of semidirect products of semigroups, though not needed in this paper, and for historical background, the interested reader is referred to the work of Nico [27] and Preston [28], [29], [30], [31].

We now describe a construction which encompasses both semidirect and reverse semidirect products, and which arises naturally in the decomposition theory we develop later for circle subgroups and submonoids of rings. The notation is due to Rosenmai [36]. Suppose that we have monoids M and N and maps

⊲ : M × N ⟶ M , (m, n) ↦ m ⊲ n
⊳ : M × N ⟶ N , (m, n) ↦ m ⊳ n

which satisfy the following conditions, known as the general product axioms: (P1) (∀m ∈ M)(∀n_{1} , n_{2} ∈ N) m ⊲ (n_{1}n_{2}) = (m ⊲ n_{1}) ⊲ n_{2} (P2) (∀m_{1} , m_{2} ∈ M)(∀n ∈ N) (m_{1}m_{2}) ⊳ n = m_{1} ⊳ (m_{2} ⊳ n)

(P3) (∀m_{1} , m_{2} ∈ M)(∀n ∈ N) (m_{1}m_{2}) ⊲ n = (m_{1} ⊲ (m_{2} ⊳ n))(m_{2} ⊲ n) (P4) (∀m ∈ M)(∀n_{1} , n_{2} ∈ N) m ⊳ (n_{1}n_{2}) = (m ⊳ n_{1})((m ⊲ n_{1}) ⊳ n_{2}) (P5) (∀m ∈ M) m ⊲ 1_{N} = m (P6) (∀n ∈ N) 1_{M} ⊳ n = n (P7) (∀n ∈ N) 1_{M} ⊲ n = 1_{M} (P8) (∀m ∈ M) m ⊳ 1_{N} = 1_{N}

Now form the (external) general product

N ⊛ M = { (n, m) | n ∈ N , m ∈ M }

with multiplication

(n_{1}, m_{1})(n_{2}, m_{2}) = (n_{1}(m_{1} ⊳ n_{2}), (m_{1} ⊲ n_{2})m_{2})

which may be routinely seen to form a monoid with identity element (1, 1) .

If m ⊲ n = m for all m ∈ M , n ∈ N then one may check that this reduces to the semidirect product

N ⊛ M = N ⋊_{φ} M (1.3)

where mφ : n ↦ m ⊳ n for m ∈ M , n ∈ N . If m ⊳ n = n for all m ∈ M , n ∈ N then this reduces to the reverse semidirect product

N ⊛ M = N ⋉_{ψ} M (1.4)

where nψ : m ↦ m ⊲ n for m ∈ M , n ∈ N . If M and N are groups then one may check that N ⊛ M is also a group and, for m ∈ M , n ∈ N ,

(n, m)^{−1} = (m^{−1} ⊳ n^{−1}, m^{−1} ⊲ n^{−1}) . (1.5)

The concept of a general product was first studied for groups by B.H. Neumann [26], and subsequently by Zappa [41] and Casadio [2]. For further development in the theory of groups the reader is referred also to the work of Rédei and Szép [32], [33], [34], [35], [38], who introduce the term skew product. The concept for semigroups and monoids has been developed by Kunze [14], [15], [16], [17], who refers to them as bilateral semidirect products, focusing attention on transformation semigroups and applications to automata theory. The terminology that we use has been popularized by Lavers [18], [19] who finds applications in the theory of vine monoids and monoid presentations. We remark that axioms (P1), (P2), (P3), (P4) define a semigroup general product, though we have no use for this wider notion in this paper. One may ask whether there is a simple criterion for recognizing when a monoid is isomorphic to the general product of two of its submonoids. Call a monoid M an internal general product of submonoids N_{1} and N_{2} if M = N_{1}N_{2} (monoid product of sets) and factorizations are unique, that is

(∀m ∈ M)(∃! n_{1} ∈ N_{1})(∃! n_{2} ∈ N_{2}) m = n_{1}n_{2} .

It is straightforward to verify the following result, first noted by Kunze [14].

Proposition 1.1. If a monoid M is the internal general product of submonoids N_{1} and N_{2} then

M ≅ N_{1} ⊛ N_{2}

under the map n_{1}n_{2} ↦ (n_{1} , n_{2}) for n_{1} ∈ N_{1} , n_{2} ∈ N_{2} , with respect to the mappings ⊲ and ⊳ defined by the equation

n_{2}n_{1} = (n_{2} ⊳ n_{1})(n_{2} ⊲ n_{1})

for unique n_{2} ⊳ n_{1} ∈ N_{1} and n_{2} ⊲ n_{1} ∈ N_{2} .

Call a monoid M with submonoids N_{1}, N_{2} an internal semidirect [reverse semidirect] product of N_{1} by N_{2} if M is an internal general product of N_{1} and N_{2} [ N_{2} and N_{1}] and

(∀n_{1} ∈ N_{1})(∀n_{2} ∈ N_{2})(∃n_{1}^{∗} ∈ N_{1}) n_{2}n_{1} = n_{1}^{∗}n_{2} [ n_{1}n_{2} = n_{2}n_{1}^{∗} ] .

We deduce easily the following.

Proposition 1.2. If a monoid M is the internal semidirect [reverse semidirect] product of N_{1} by N_{2} then

M ≅ N_{1} ⋊_{ϕ} N_{2} [ N_{2} ⋉_{ϕ} N_{1} ]

where ϕ : N_{2} → End (N_{1}) is defined by the equation

n_{2}n_{1} = (n_{1}(n_{2}ϕ))n_{2} [ n_{1}n_{2} = n_{2}(n_{1}(n_{2}ϕ)) ]

for n_{1} ∈ N_{1}, n_{2} ∈ N_{2}.

2. Examples

We give some contrasting examples using groups and monoids illustrating general, semidirect and reverse semidirect products. The group examples will be revisited, from a different direction, in Section 8, as an application of the theory of presentations which we develop in Section 7.

Example 2.1. We give a simple example of a general product which is neither semidirect nor reverse semidirect. Let M = { x^{i} | i ∈ ℤ^{+} ∪ {0} } be the infinite monogenic monoid and define, for i , j ∈ ℤ^{+} ∪ {0} ,

x^{i} ⊲ x^{j} = \begin{cases}1 & if j ≥ i \\ x^{i−j} & if i > j \end{cases}  ,  x^{i} ⊳ x^{j} = \begin{cases}1 & if i ≥ j \\ x^{j−i} & if j > i . \end{cases}

Then it is routine to check that the general product axioms are satisfied, so we may form the general product M ⊛ M , and further that

M ⊛ M ≅ ⟨ a , b | ab = 1 ⟩_{monoid} ,

the bicyclic monoid [9, Example V.4.6], [6, Section 1.12].

We give two examples of general products of groups which we will see later arise as the circle groups of the ring of 2 × 2 matrices over pℤ_{p^{3}} where p is an odd and even prime respectively.

Example 2.2. Let p be any prime and

G = ⟨ x , y | x^{p^{2}} = y^{p^{2}} = 1 , x^{y} = x^{1−p} ⟩ .

Observe that z ↦ z^{1−p} is an automorphism of C_{p^{2}} of order p , with respect to which we may form the semidirect product C_{p^{2}} ⋊ C_{p^{2}} , and this is isomorphic to G . Thus we may write

G = { x^{i}y^{j} | i , j ∈ ℤ_{p^{2}} }

with multiplication

x^{i_{1}}y^{j_{1}}x^{i_{2}}y^{j_{2}} = x^{i_{1}+i_{2}(1+p)^{j_{1}}}y^{j_{1}+j_{2}} .

Now define ⊲ , ⊳ : G × G ⟶ G by the rules

x^{i}y^{j} ⊲ x^{k}y^{l} = x^{i(1−p)^{−l}}y^{j−ikp} , x^{i}y^{j} ⊳ x^{k}y^{l} = x^{k(1−p)^{j}}y^{l+ikp} ,

interpreting the expressions in the exponents always as elements of ℤ_{p^{2}}. The verification of axioms (P5), (P6), (P7), (P8) is trivial and (P1), (P2) straightforward. To check (P3) note that, for z ∈ C_{p^{2}} ,

z^{(1±p)^{p}} = z , (z^{p})^{(1±p)} = z^{p} .
Then  [x^{i_{1}}y^{j_{1}} ⊲ (x^{i_{2}}y^{j_{2}} ⊳ x^{k}y^{l})](x^{i_{2}}y^{j_{2}} ⊲ x^{k}y^{l})
= x^{i_{1}(1−p)^{−l−i_{2}kp}+i_{2}(1−p)^{−l}(1+p)^{j_{1}−i_{1}k(1−p)^{j_{2}} p}}y^{j_{1}−i_{1}k(1−p)^{j_{2}}p+j_{2}−i_{2}kp}
= x^{i_{1}(1−p)^{−l}+i_{2}(1−p)^{−l}(1+p)^{j_{1}}}y^{j_{1}−i_{1}kp+j_{2}−i_{2}kp}
= x^{(i_{1}+i_{2}(1−p)^{j_{1}})(1+p)^{−l}}y^{j_{1}+j_{2}−(i_{1}+i_{2}(1+p)^{j_{1}})kp}
= (x^{i_{1}}y^{j_{1}}x^{i_{2}}y^{j_{2}}) ⊲ x^{k}y^{l} ,

which verifies (P3). The verification of (P4) is similar. Thus we may form the general product G ⊛ G. Observe that

y^{−1} ⊳ x = x^{1+p} , y^{−1} ⊲ x = y^{−1} , x ⊳ y = y , x ⊲ y = x^{1+p} ,
y ⊳ y = y ⊲ y = y , x ⊳ x = xy^{p} , x ⊲ x = xy^{−p} .

It follows, by an obvious identification of generators and a straightforward counting argument (using the previous observations to check satisfiability of the relations below), that G ⊛ G is isomorphic to the group

⟨ x_{1}, y_{1}, x_{2}, y_{2} | x_{i}^{p^{2}} = y_{i}^{p^{2}} = 1 , x_{i}^{y_{i}} = x_{i}^{1−p} (∀i) , x_{i}^{y_{j}} = x_{i}^{1+p} (∀i ≠ j)
[y_{1} , y_{2}] = 1 , [x_{1} , x_{2}] = y_{1}^{−p}y_{2}^{p} ⟩ .

Example 2.3. Consider

H = ⟨ x, y, z | x^{4} = y^{2} = z^{2} = 1 , [x, y] = [y, z] = 1 , x^{z} = x^{−1} ⟩

which may be viewed as a semidirect product, in at least two ways, isomorphic to

C_{4} ⋊ (C_{2} × C_{2}) or (C_{4} × C_{2}) ⋊ C_{2}

where the copy of C_{4} and the second copy of C_{2} form a dihedral subgroup of order 8 . We may write

H = { x^{i}y^{j}z^{k} | i ∈ ℤ_{4} , j, k ∈ ℤ_{2} }

with multiplication

x^{i_{1}}y^{j_{1}}z^{k_{1}}x^{i_{2}}y^{j_{2}}z^{k_{2}} = x^{i_{1}+i_{2}(−1)^{k_{1}}}y^{j_{1}+j_{2}}z^{k_{1}+k_{2}} .

Now define ⊲ , ⊳ : H × H ⟶ H by the rules

x^{i}y^{j}z^{k} ⊲ x^{l}y^{m}z^{n} = x^{(−1)^{n}i}y^{j+il}z^{k} , x^{i}y^{j}z^{k} ⊳ x^{l}y^{m}z^{n} = x^{(−1)^{k}l}y^{m+il}z^{n} .

It is straightforward to verify the general product axioms (relying on the fact that y = y^{−1} for (P3)). Thus we may form the general product H ⊛ H which, by a straightforward counting argument, is isomorphic to

⟨ x_{1} , y_{1} , z_{1} , x_{2} , y_{2} , z_{2} | x_{i}^{4} = y_{i}^{2} = z_{i}^{2} = 1 , [x_{i} , y_{j}] = [y_{i} , z_{j}] = 1 ,
x_{i}^{z_{j}} = x_{i}^{−1} , i, j = 1, 2 , [y_{1} , y_{2}] = [z_{1} , z_{2}] = 1 , [x_{1} , x_{2}] = y_{1}y_{2} ⟩ .

The differences between semidirect and reverse semidirect products become apparent when one moves beyond the class of groups. We combine both in the example below. A Munn ring M(S; P), where S is a ring and P is an m × n matrix over S^{1}, consists of n × m matrices over S with usual addition of matrices and multiplication · defined by

α · β = αP β

for α, β ∈ M(S; P), where juxtaposition denotes normal matrix multiplication. For a detailed analysis of the circle monoids of Munn rings the interested reader is referred to another paper [8] of the authors. The terminology Munn ring is due to McAlister [20], which in turn derives from the notion of Munn algebra (see [24] and [6, Section 5.2]), though in our definition above we allow an unrestricted sandwich matrix P (see also [25]).

Example 2.4. Consider the commutative monoid

M_{1} = ⟨ x , y | x^{2} = 1 , y^{3} = y^{2} , y = xy = yx ⟩_{monoid}

which is an ideal extension (in the sense of [6, Section 4.4]) of a two element null semigroup by a copy of C_{2} with zero adjoined, and we may write

M_{1} = { 1 , x , y , y^{2} = 0 } .

Then M_{1} ≅ (ℤ_{4} , ·) ≅ (ℤ_{4} , ∘) . We write C_{4} = ⟨z⟩ and induce endomorphisms xφ , yφ of C_{4} by the rules

xφ : z ↦ z^{−1} , yφ : z ↦ z^{2} .

The relations of M_{1} are satisfied in End (C_{4}) when x , y are replaced by xφ , yφ respectively, so we induce a homomorphism (= antihomomorphism, since M_{1} is commutative) φ : M_{1} ⟶ End (C_{4}) with respect to which we may form the semidirect product

M_{2} = C_{4} ⋊_{φ} M_{1} .

Clearly

M_{2} ≅ ⟨ x , y , z | relations of M_{1} , z^{4} = 1 , xz = z^{3}x , yz = z^{2}y ⟩_{monoid}

and we may write, without causing confusion,

M_{2} = { z^{i}x^{j} , z^{i}y^{k} | i ∈ ℤ_{4} , j ∈ ℤ_{2} , k ∈ {1 , 2} } .

It is not difficult to see, by a simple counting argument, that M_{2} is isomorphic to the circle monoid of the Munn ring M(ℤ_{4} ; ( \begin{matrix}1 \\ 0 \end{matrix} )) . Now put

K = ⟨ u , v | u^{4} = v^{4} = [u , v] = 1 ⟩ ≅ C_{4} × C_{4}

and induce endomorphisms xψ , yψ , zψ of K by the rules

xψ : u ↦ u^{−1} , v ↦ v
yψ : u ↦ u^{2} ,  v ↦ v
zψ : u ↦ uv^{−1} , v ↦ v .

The relations of M_{2} are satisfied in End (K) where x , y , z are replaced by xψ , yψ , zψ respectively, so we induce a homomorphism ψ : M_{2} ⟶ End (K) with respect to which we may form the reverse semidirect product

M_{3} = M_{2} ⋉_{ψ} K .

It is not difficult to verify that M_{3} is isomorphic to the circle monoid of the Munn ring M(ℤ_{4} ; ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} )) , and further that

M_{3} ≅ ⟨ x , y , z , u , v | relations of M_{2} and K , ux = xu^{3} , uy = yu^{2} ,

uz = zuv^{3} , vx = xv , vy = yv , vz = zv ⟩_{monoid} .

3. Some technical lemmas

In this section we collect together some observations of a technical nature which will be useful later in applying Tietze transformations. The proofs of Lemmas 3.1 and 3.2 are straightforward inductions and left to the reader.

64 C. Coleman, D. Easdown: Decomposition of Rings under the Circle Operation

Lemma 3.1. If G is a group and x, y, z ∈ G such that [x , y] = z and [x , z] = [y , z] = 1 then [x^{λ} , y^{μ}] = z^{λμ} for all λ , μ ∈ ℤ^{+} . Lemma 3.2. If G is a group and x, y ∈ G such that [x , y] = y^{α} for some α ∈ ℤ then

[x^{λ} , y^{μ}] = y^{μ(1−(1−α)^{λ})}

for all λ , μ ∈ ℤ^{+} . Lemma 3.3. Suppose that G is a group and x, y, z ∈ G such that [x, z] = z^{α} for some α ∈ ℤ, [y, z] = z^{2} and [x, y] = 1. Then

[x^{λ}y, z^{μ}] = z^{μ(1+(1−α)^{λ})}

for all λ, μ ∈ ℤ^{+}. Proof. Observe that z^{y} = z^{−1}, so, by Lemma 3.2,

[x^{λ}y, z^{μ}] = [x^{λ}, z^{μ}]^{y} [y, z^{μ}] = z^{−μ(1−(1−α)^{λ})}z^{2μ} = z^{μ(1+(1−α)^{λ})} . □

Lemma 3.4. Let p be a prime, t ≥ 3, and put

q = \{ \begin{matrix}p & if p ≠ 2 \\ 4 & if p = 2 . \end{matrix}

Suppose G is a group, x, y, z, w ∈ G such that x, y, z, w each have order dividing p^{t} ,

x^{z} = x^{1−q} , x^{w} = x^{1−q′} , y^{z} = y^{1−q′} , y^{w} = y^{1−q} , [z , w] = 1

(all quasi-inversion taking place in ℤ_{p^{t}}), and for each m = 0 , …, p^{t−3} − 1 ,

x^{1−(−mp^{2})′}y = z^{−α}yx^{1−(−mp^{2})′}w^{α}

where α is the least positive integer such that

(1 − q)^{α} = 1 + (1 − (−mp^{2})′)p^{2}

in ℤ_{p^{t}} (which exists because qℤ_{p^{t}} = ⟨q⟩_{∘}). Then, for all λ , μ ∈ ℤ^{+} ,

x^{λ}y^{μ} = z^{−ν}y^{μ}x^{λ}w^{ν}

where ν is the least positive integer such that

(1 − q)^{ν} = 1 + λμp^{2}

in ℤ_{p^{t}} .

Proof. The case λ = μ = 1 is covered by the hypothesis (when m = 0), which starts an induction. In the following, since orders divide p^{t} , we may interpret exponents as elements of ℤ_{p^{t}} . Let λ > 1 . By an inductive hypothesis, choosing α so that (1 − q)^{α} = 1 + (λ − 1)p^{2},

x^{λ}y = xx^{λ−1}y = xz^{−α}yx^{λ−1}w^{α}

= z^{−α}x^{z^{−α}}yx^{λ−1}w^{α}

= z^{−α}x^{(1−q)^{−α}}yx^{λ−1}w^{α}

= z^{−α}z^{−β}yx^{(1−q)^{−α}}w^{β}x^{λ−1}w^{α} ,

choosing β such that (1 − q)^{β} = 1 + (1 − q)^{−α}p^{2} by the hypothesis, since (1 − q)^{−α} = 1 − (−(λ − 1)p^{2})′, so that

x^{λ}y = z^{−(α+β)}yx^{(1−q)^{−α}}(x^{λ−1})^{w^{−β}}w^{β}w^{α}

= z^{−(α+β)}yx^{(1−q)^{−α}}x^{(1−q′)^{−β}(λ−1)}w^{α+β}

= z^{−δ}yx^{λ}w^{δ}

where δ = α + β, after observing that (performing arithmetic in ℤ_{p^{t}})

(1 − q)^{−α} + (1 − q′)^{−β}(λ − 1) = (1 − q)^{−α} + (1 − q)^{β}(λ − 1)

= (1 − q)^{−α} + (1 + (1 − q)^{−α}p^{2})(λ − 1) = λ − 1 + (1 − q)^{−α}(1 + (λ − 1)p^{2}) = λ − 1 + 1 = λ .

Further we have that

(1 − q)^{δ} = (1 − q)^{α}(1 − q)^{β}

= (1 − q)^{α}(1 + (1 − q)^{−α}p^{2}) = (1 − q)^{α} + p^{2} = 1 + λp^{2} .

Now let μ > 1 , λ ≥ 1 . By an inductive hypothesis, we have, choosing γ such that (1 − q)^{γ} = 1 + λ(μ − 1)p^{2} ,

x^{λ}y^{μ} = x^{λ}y^{μ−1}y = z^{−γ}y^{μ−1}x^{λ}w^{γ}y

= z^{−γ}y^{μ−1}w^{γ}(x^{λ})^{w^{γ}}y

= z^{−γ}y^{μ−1}w^{γ}x^{(1−q′)^{γ}λ}y

= z^{−γ}y^{μ−1}w^{γ}z^{−ϵ}yx^{(1−q′)^{γ}λ}w^{ϵ} ,

choosing ϵ such that (1 − q)^{ϵ} = 1 + (1 − q′)^{γ}λp^{2} by the first half, so that, since [z , w] = 1 ,

x^{λ}y^{μ} = z^{−γ}y^{μ−1}z^{−ϵ}w^{γ}yx^{(1−q′)^{γ}λ}w^{ϵ}

= z^{−γ}z^{−ϵ}(y^{μ−1})^{z^{−ϵ}}y^{w^{−γ}}w^{γ}x^{(1−q′)^{γ}λ}w^{ϵ}

= z^{−(γ+ϵ)}y^{(1−q′)^{−ϵ}(μ−1)}y^{(1−q)^{−γ}}(x^{(1−q′)^{γ}λ})^{w^{−γ}}w^{γ}w^{ϵ}

= z^{−σ}y^{(1−q′)^{−ϵ}(μ−1)+(1−q)^{−γ}}x^{(1−q′)^{−γ}(1−q′)^{γ}λ}w^{σ}

= z^{−σ}y^{μ}x^{λ}w^{σ} where σ = ϵ + γ, after observing that

(1 − q′)^{−ϵ}(μ − 1) + (1 − q)^{−γ} = (1 − q)^{ϵ}(μ − 1) + (1 − q)^{−γ}
= (1 + (1 − q′)^{γ}λp^{2})(μ − 1) + (1 − q)^{−γ}
= μ − 1 + (1 − q)^{−γ}(λ(μ − 1)p^{2} + 1)
= μ − 1 + 1 = μ .

Further we have that

(1 − q)^{σ} = (1 − q)^{ϵ}(1 − q)^{γ}
= (1 + (1 − q′)^{γ}λp^{2})(1 − q)^{γ}
= (1 − q)^{γ} + λp^{2}
= 1 + λ(μ − 1)p^{2} + λp^{2}
= 1 + λμp^{2} . □

The next result is used in developing the presentation in Section 6 for circle groups of rings of matrices over radical rings. Though we only apply it in this paper in a group-theoretic context, it is no harder to state and prove for monoids, and it is useful in studying the circle monoids of Munn rings (see [8]). Note that the angular brackets refer to submonoid generation for the remainder of this section.

Lemma 3.5. Let M be a monoid and n a positive integer. For each i , j ∈ { 1, …, n }, let X_{ij} ⊆ M and put Y_{ij} = ⟨ X_{ij} ⟩ . Suppose that

(1) M = ⟨ ∪_{i,j} X_{ij} ⟩ .

(2) ( ∀i ≠ l , j ≠ k)( ∀x ∈ X_{ij})( ∀y ∈ X_{kl}) xy = yx

(3) ( ∀i, j , k ≠ i)( ∀x ∈ X_{ij})( ∀y ∈ X_{jk})( ∃z_{1}, z_{2}, w_{1}, w_{2} ∈ Y_{ik})

xy = z_{1}yx = yxz_{2} , yx = xyw_{1} = w_{2}xy ;

(4) ( ∀i > j)( ∀x ∈ Y_{ij})( ∀y ∈ Y_{ji})( ∃z ∈ Y_{jj})( ∃w ∈ Y_{ii}) xy = zyxw .

Then M = ∏_{i=1}^{n} ∏_{j=1}^{n} Y_{ij} , so, in particular, if M is finite, |M| ≤ ∏_{i=1}^{n} ∏_{j=1}^{n} |Y_{ij}| .

We prove Lemma 3.5 by first developing a sequence of lemmas, each of which is assumed to have the hypotheses of Lemma 3.5.

Lemma 3.6. ( ∀j ≠ i)( ∀x ∈ Y_{ii})( ∀y ∈ Y_{ij} [ Y_{ji}] )( ∃z , w ∈ Y_{ij} [ Y_{ji}] )

yx = xz  and  xy = wx .

Proof. This follows by (3) and a simple induction on the number of generators. □

Lemma 3.7. ( ∀i ≠ j ≠ k ≠ i)( ∀x ∈ Y_{jk})( ∀y ∈ Y_{ij})( ∃z_{1}, z_{2} ∈ Y_{ik})

yx = xyz_{1}  and  xy = yxz_{2} .

Proof. Suppose i ≠ j ≠ k ≠ i . By (2), elements of Y_{ik} commute with elements of Y_{ij} ∪ Y_{jk} , so, by a simple induction on the number of generators, it suffices to suppose x ∈ X_{jk} , y ∈ X_{ij} , and then the result follows immediately by (3). □

For i ∈ {1 , …, n} , put

R_{i} = Y_{i1} … Y_{in} .

Lemma 3.8. For each i ∈ {1, …, n},

R_{i} = ⟨ ∪^{n}_{j=1}X_{ij} ⟩ ,

so, in particular, R_{i}R_{i} = R_{i}.

Proof. Clearly ∪^{n}_{j=1}X_{ij} ⊆ R_{i} ⊆ ⟨ ∪^{n}_{j=1}X_{ij} ⟩ , so to prove the Lemma it suffices to show R_{i} is

closed under multiplication on the right by elements of ∪^{n}_{j=1}X_{ij} . Let g = y_{1} … y_{n} ∈ R_{i} where

y_{j} ∈ Y_{ij} for j = 1 , …, n . Let k ∈ { 1 , …, n } and choose x ∈ X_{ik} . We show gx ∈ R_{i} . If k > i then, by (2),

gx = y_{1} … y_{k−1}( y_{k} x)y_{k+1} … y_{n} ∈ R_{i} .

If k = i then, by Lemma 3.6, for each j > i , y_{j}x = xz_{j} for some z_{j} ∈ Y_{ij} , so

gx = y_{1} … y_{i−1}( y_{i} x)z_{i+1} … z_{n} ∈ R_{i} .

If k < i then, by (2) and Lemma 3.6, there exists z ∈ Y_{ik} such that

gx = y_{1} … y_{i} x y_{i+1} … y_{n} = y_{1} … y_{i−1} z y_{i} y_{i+1} … y_{n}

= y_{1} … y_{k−1}( y_{k} z)y_{k+1} … y_{n} ∈ R_{i} . □

Lemma 3.9. ( ∀i > j)( ∀k) R_{i}Y_{jk} ⊆ R_{j}R_{i} .

Proof. Suppose i , j , k ∈ { 1, …, n } and j < i . Let g ∈ R_{i} , x ∈ X_{jk} , so g = y_{1} … y_{n} for some y_{1} ∈ Y_{i1} , …, y_{n} ∈ Y_{in} . If i ≠ k then, by (2),

gx = y_{1} … y_{j} x y_{j+1} … y_{n} = y_{1} … y_{j−1} x w y_{j+1} … y_{n}

for some w ∈ Y_{ij}, by Lemma 3.6, if k = j, and for w = y_{j}z for some z ∈ Y_{ik}, by Lemma 3.7, if k ≠ j, so that, by (2) and Lemma 3.8,

gx = x( y_{1} … y_{j−1} w y_{j+1} … y_{n})

∈ X_{jk}⟨ ∪^{n}_{l=1}X_{il} ⟩ = X_{jk}R_{i} ⊆ R_{j}R_{i} .

If i = k then, making free use of (2) throughout,

gx = y_{1} … y_{i} x( y_{i+1} z_{i+1} ) … ( y_{n} z_{n})

( ∃z_{i+1} ∈ Y_{j,i+1} ) … ( ∃z_{n} ∈ Y_{jn}) by Lemma 3.7

= y_{1} … y_{i−1} w y_{i} y_{i+1} … y_{n} z_{i+1} … z_{n}
( ∃w ∈ Y_{ji}) by Lemma 3.6
= y_{1} … y_{j} w ( y_{j+1} z_{j+1} ) … ( y_{i−1} z_{i−1} )y_{i} … y_{n} z_{i+1} … z_{n}

( ∃z_{j+1} ∈ Y_{j,j+1} ) … ( ∃z_{i−1} ∈ Y_{j,i−1} ) by Lemma 3.7

= y_{1} … y_{j} w y_{j+1} … y_{n} z_{j+1} … z_{i−1} z_{i+1} … z_{n}
= y_{1} … y_{j−1} ( u w y_{j} v) y_{j+1} … y_{n} z_{j+1} … z_{i−1} z_{i+1} … z_{n}
( ∃u ∈ Y_{jj})( ∃v ∈ Y_{ii}) by (4)
= u y_{1} … y_{j−1} w y_{j} v y_{j+1} … y_{n} z_{j+1} … z_{i−1} z_{i+1} … z_{n}
= u w ( y_{1} z_{1} ) … ( y_{j−1} z_{j−1} ) y_{j} v y_{j+1} … y_{n} z_{j+1} … z_{i−1} z_{i+1} … z_{n}

( ∃z_{1} ∈ Y_{j1} ) … ( ∃z_{j−1} ∈ Y_{j,j−1} ) by Lemma 3.7

= ( u w z_{1} … z_{j−1} )( y_{1} … y_{j} v y_{j+1} … y_{n})( z_{j+1} … z_{i−1} z_{i+1} … z_{n})
∈ R_{j}R_{i}( z_{j+1} … z_{i−1} z_{i+1} … z_{n}) ⊆ R_{j}R_{j}R_{i} = R_{j}R_{i} ,

in the last line, by iterating the previous case (when i ≠ k), and also by Lemma 3.8. This proves R_{i}X_{jk} ⊆ R_{j}R_{i} . It follows immediately that R_{i}Y_{jk} ⊆ R_{j}R_{i} . □

Lemma 3.10. ( ∀i > j) R_{i}R_{j} ⊆ R_{j}R_{i} .

Proof. This follows immediately by Lemmas 3.8 and 3.9. □

Proof of Lemma 3.4. We have to show M = R_{1} … R_{n} . Clearly ⋃_{i,j}X_{ij} ⊆ R_{1} … R_{n} , so it

suffices to show R_{1} … R_{n} is closed under multiplication on the right by elements of ⋃_{i,j}X_{ij} .

For any j,

R_{n}X_{nj} ⊆ ⟨ ∪^{n}_{k=1}X_{nk} ⟩ = R_{n} ,

by Lemma 3.8, so that

R_{1} … R_{n}X_{nj} ⊆ R_{1} … R_{n} ,

and, for any i < n,

R_{1} … R_{n}X_{ij} ⊆ R_{1} … R_{n}R_{i} ⊆ ( R_{1} … R_{i})( R_{i} … R_{n}) = R_{1} … R_{n} ,

since ( R_{i+1} … R_{n})R_{i} ⊆ R_{i} … R_{n}, by Lemma 3.10, and since R_{i}R_{i} = R_{i}, by Lemma 3.8. This completes the proof of Lemma 3.5. □

4. Representations and antirepresentations

Consider a ring S. In what follows we develop a sequence of steps leading to naturally occurring representations and antirepresentations of circle submonoids of S by endomorphisms (or automorphisms if the submonoid is a subgroup) of additive subgroups of (S , +) . From these we may form external semidirect and reverse semidirect products. In the next section we will find conditions under which these become internal, leading to a decomposition theory for a large class of circle monoids and groups.

(1) Define

ρ_{S} , λ_{S} : S ⟶ End (S , +)

by, for x, y ∈ S,

xρ_{S} : y ↦ yx , xλ_{S} : y ↦ xy .

It is well known (and easily checked) that ρ_{S} and λ_{S} are a representation and antirepresentation respectively of S , and faithful if S has 1 .

(2) Let M be a multiplicatively closed subset of S^{1} and T , U be additive subgroups of S^{1} closed under multiplication by elements of M on the right, left respectively. Define

ρ_{M,T} : M ⟶ End (T , +) by mρ_{M,T} : t ↦ tm (m ∈ M , t ∈ T)

and

λ_{M,U} : M ⟶ End (U , +) by mλ_{M,U} : u ↦ mu (m ∈ M , u ∈ U) .

Then ρ_{M,T} and λ_{M,U} are a representation and antirepresentation respectively, resulting from ρ_{S^{1}} and λ_{S^{1}} by restriction. Further, it is easy to see that if M ≤ G(S^{1}) , then

(4.1) ρ_{M,T} : M ⟶ Aut (T , +) and λ_{M,U} : M ⟶ Aut (U , +) .

(3) Let M be a subset of S closed under ∘ , and T , U be additive subgroups of S^{1} closed under ordinary ring multiplication by elements of M (and hence also by elements of M̂) on the right, left respectively. Define the composites

ρ̂_{M,T} = ∘ ρ_{M,T} and λ̂_{M,U} = ∘ λ_{M,U} ,

so

mρ̂_{M,T} : t ↦ tm̂ = t − tm (m ∈ M , t ∈ T)

and

mλ̂_{M,U} : u ↦ m̂u = u − mu (m ∈ M , u ∈ U) .

Because they are composites with a monoid homomorphism, we have that ρ̂_{M,T} and λ̂_{M,U} are a representation and antirepresentation respectively. Further, by (4.1) , if M ≤ (G(S) , ∘) then

(4.2)  ρ̂_{M,T} : M ⟶ Aut (T , +) and λ̂_{M,U} : M ⟶ Aut (U , +) .

(4) Suppose, in addition to the hypothesis of (3), that there is an anti-isomorphism^{†} : M ⟶ M (for example ^{†} might be quasi-inversion if M ≤ (G(S) , ∘)). Define the composites

ρ̂ †_{M,T} = ^{†} ∘ ρ̂_{M,T} and λ̂ †_{M,U} = ^{†} ∘ λ̂_{M,U}

so

mρ̂ †_{M,T} : t ↦ t − tm^{†} (m ∈ M , t ∈ T)

and

mλ̂ †_{M,U} : u ↦ u − m^{†}u (m ∈ M , u ∈ U) .

Because they are composites with an anti-isomorphism, ρ̂ †_{M,T} and λ̂ †_{M,U} are an antirepresentation and representation respectively. Further, by (4.2), if M ≤ (G(S) , ∘) then

ρ̂ †_{M,T} : M ⟶ Aut (T , +) and λ̂ †_{M,U} : M ⟶ Aut (U , +) .

As a result of these four steps, we may, under the appropriate hypotheses, form the external semidirect products

U ⋊λ̂_{M,U} M and T ⋊_{ρ_{M,T}^{†}} M ,

and the external reverse semidirect products

M ⋉ρ̂_{M,T} T and M ⋉λ̂_{M,U}^{†} U .

In the case that M ≤ (G(S), ∘), and ^{†} is quasi-inversion, then all of these are groups and, by (1.2),

M ⋉ρ̂_{M,T} T ≅ T ⋊_{ρ_{M,T}^{†}} M

and

U ⋊λ̂_{M,U} M ≅ M ⋉λ̂_{M,U}^{†} U .

5. Circle Decompositions

In this section we find decompositions of circle monoids and groups using internal general, semidirect and reverse semidirect products, and, in particular, look for conditions under which the external constructions of the previous section can be realized up to isomorphism. We begin with general conditions under which additive and circle decompositions coincide and the circle factorization is unique.

Lemma 5.1. Suppose (I , +) ≤ (S , +) , (H , ∘) ≤ (G(S) , ∘) and I ∩ ⟨ H⟩_{+} = {0} . If I absorbs multiplication on the right [ left ] by elements of H then

I + H = I ∘ H [ H ∘ I]

and circle factorizations are unique.

Proof. Suppose I absorbs multiplication on the right by elements of H . If x ∈ I and h ∈ H then xĥ , x\overline{h} ∈ I ,

x ∘ h = x + h − xh = xĥ + h ∈ I + H

and

x + h = x\overline{h} + h − x\overline{h}h = (x\overline{h}) ∘ h ∈ I ∘ H .

This proves I + H = I ∘ H . If x_{1} , x_{2} ∈ I , h_{1} , h_{2} ∈ H and x_{1} ∘ h_{1} = x_{2} ∘ h_{2} then

h_{1} − h_{2} = x_{2} − x_{1} + x_{1}h_{1} − x_{2}h_{2} ∈ I ∩ ⟨ H⟩_{+} = {0} ,

so h_{1} = h_{2} and x_{1} = x_{1} ∘ h_{1} ∘ h_{1}′ = x_{2} ∘ h_{1} ∘ h_{1}′ = x_{2} . This proves circle factorizations are unique. The other half of the lemma is dual. □

Theorem 5.2. Suppose that I is a subring of S, (H , ∘) ≤ (G(S) , ∘) , I ∩ ⟨ H⟩_{+} = {0} and I absorbs multiplication by elements of H on both the right and left. Then

I + H = I ∘ H = H ∘ I

and I + H is the internal semidirect product of (I , ∘) by (H , ∘) . Furthermore

I + H ≅ (I , ∘) ⋊_{θ} (H , ∘)

where θ is defined by

hθ : x ↦ ĥx\overline{h} (x ∈ I , h ∈ H) .

Proof. Observe that I + H is a submonoid of (S , ∘) , by the formula

(x_{1} + h_{1}) ∘ (x_{2} + h_{2}) = (x_{1} ∘ x_{2}) + (h_{1} ∘ h_{2}) − x_{1}h_{2} − h_{1}x_{2} (5.1)

and the fact that I absorbs multiplication by elements of H on both the right and the left, and, by Lemma 5.1, that I + H = I ∘ H = H ∘ I and circle factorizations are unique. If x ∈ I , h ∈ H then h′ ∘ x ∘ h = x − h′x − xh + h′xh ∈ I so that I is closed under conjugation by elements of H . It follows immediately that I + H is the internal semidirect product of (I , ∘) by (H , ∘) . The last claim follows easily by observing, for x ∈ I , h ∈ H , that

h ∘ x = h + x − hx = ĥx + h = (ĥx\overline{h}) ∘ h . □

Corollary 5.3. If I is a subring of S , (H , ∘) ≤ (G(S) , ∘) , I ∩ ⟨ H⟩_{+} = {0} , I absorbs multiplication by elements of H on the right [ left ] and H annihilates I by multiplication on the left [ right ], then

I + H = I ∘ H = H ∘ I ,

I + H is the internal semidirect product of (I , ∘) by (H , ∘), and

I + H ≅ (I , ∘) ⋊_{ρ\begin{matrix}′ \\ H,I \end{matrix}} (H , ∘) [ (I , ∘) ⋊λ̂_{H,I} (H , ∘) ] .

Proof. This is immediate from Theorem 5.2, noting that for x ∈ I , h ∈ H ,

ĥx\overline{h} = \begin{cases}x\overline{h} & if hx = 0 \\ ĥx & if x\overline{h} = 0 . \end{cases} □

Theorem 5.4. Suppose that I is a subring of S , (H , +) ≤ (S, +), I ∩ H = {0}, (H, ∘) ≤ (G(S) circ) and I and H absorb each other by multiplication on the right [ left ]. Then

I + H = I ∘ H [ H ∘ I]

and I + H is the internal general product of (I , ∘) with (H , ∘) [ (H , ∘) with (I , ∘) ]. Furthermore

I + H ≅ (I , ∘) ⊛ (H , ∘) [ (H , ∘) ⊛ (I , ∘) ]

where the mappings ⊲ and ⊳ are defined by, for x ∈ I, h ∈ H,

h ⊲ x = hx̂ , h ⊳ x = x\overline{hx̂} [ x ⊲ h = \overline{x̂h}x , x ⊳ h = x̂h]

Proof. We prove the “right” half, the other being dual. Observe that I + H is a submonoid of (S , ∘) (again by equation (5.1)) so, by Lemma 5.1, I + H = I ∘ H is the internal general product of (I , ∘) with (H , ∘) . The last claim follows by observing that, for x ∈ I , h ∈ H ,

h ∘ x = x + hx̂ = (x\overline{hx̂}) ∘ (hx̂) . □

Corollary 5.5. If I is a subring of S , (H , +) ≤ (S , +) , I ∩ H = {0} , (H , ∘) ≤ (G(S) , ∘) , H absorbs elements of I by multiplication on the left [ right ] and I annihilates H by multiplication on the right [ left ], then

I + H = H ∘ I [ I ∘ H] ,

I + H is the internal semidirect [ reverse semidirect ] product of (H , ∘) by (I , ∘) and

I + H ≅ (H , ∘) ⋊λ̂_{I,H} (I , ∘) [ (I , ∘) ⋉ρ̂_{I,H} (H , ∘) ]

Proof. This is immediate from Theorem 5.4, noting that, for x ∈ I , h ∈ H ,

x = \begin{cases}\overline{x̂h}x & if (x̂h)x = 0 \\ x\overline{hx̂} & if x(hx̂) = 0 . \end{cases} □

In the applications that now follow, all of the submonoids are subgroups, and the conclusions of Corollaries 5.4 and 5.5 carry the same information (in accordance with (1.2)). In [8] the authors consider monoids which are not groups (see Example 2.4 above) and Theorems 5.2 and 5.4 and their corollaries play markedly different roles.

6. Matrices over a radical ring

Let S be a radical ring and n ≥ 1. Then S is an abelian group under addition and a (not necessarily abelian) group under circle. (Even when both groups are abelian they need not be isomorphic; for example (2ℤ_{8} , +) is cyclic of order 4 , whilst (2ℤ_{8} , ∘) is isomorphic to the Klein 4 group.) Then M_{n}(S) = J (M_{n}(S)) = G(M_{n}(S)) is a group under ∘ and has many possible decompositions. In this section we give a decomposition involving rows (which dualizes to columns) and then a contrasting decomposition involving both rows and columns leading to a recursive formula. In both cases (M_{n}(S) , ∘) is built from (S , +) and (S , ∘) using direct, semidirect and general products. All of the anti-representations involved in the use of semidirect products are described explicitly using the theory and notation of Section 5. The ⊲ , ⊳ mappings involved in forming general products, whilst not explicitly described here, can be gleaned from results in Section 5.

Put M = M_{n}(S) and for i , j ∈ { 1 , …, n } ,
X_{ij} = { α ∈ M | α_{kl} = 0 if k ≠ i or l ≠ j } ,
R_{i} = X_{i1} + … + X_{in} ,
R̃_{i} = X_{i1} + … + X_{i,i−1} + X_{i,i+1} + … + X_{in} ,
C_{i} = X_{1i} + … + X_{ni} ,
C̃_{i} = X_{1i} + … + X_{i−1,i} + X_{i+1,i} + … + X_{ni} ,
T_{i} = R_{1} + … + R_{i} ,
M_{i} = { α ∈ M | α_{kl} = 0 if k > i or l > i } .

It is straightforward to check that all of these are subrings and circle subgroups of M . We develop our understanding of (M , ∘) through the following sequence of steps.

(1) If i ≠ j then X_{ij} is both an ideal and a normal subgroup of R_{i}, and X_{ij} is a null ring (so circle coincides with addition) which annihilates elements of R_{i} , and X_{ii} in particular, by multiplication on the left. Clearly then, for each i ,

R̃_{i} = X_{i1} ∘ … ∘ X_{i,i−1} ∘ X_{i,i+1} ∘ … ∘ X_{in}

and, for j ≠ i ,

X_{ij} ∩ (∑_{\begin{matrix}k≠j \\ k≠i \end{matrix}} X_{ik}) = \{0\} ,

yielding an internal direct product decomposition of R̃_{i} , whence

(R̃_{i} , ∘) ≅ (S , +)^{n−1} . (6.1)

(2) For each i , (X_{ii} , ∘) ≅ (S , ∘) and X_{ii} is a left ideal of R_{i} . Further, R̃_{i} absorbs multiplication by elements of X_{ii} on the left and is annihilated by X_{ii} by multiplication on the

74 C. Coleman, D. Easdown: Decomposition of Rings under the Circle Operation

right. Also R̃_{i} ∩ X_{ii} = {0} . Hence, by Corollary 5.3 or 5.5 and isomorphism (6.1)

R_{i} = R̃_{i} + X_{ii} = R̃_{i} ∘ X_{ii}
≅ R̃_{i} ⋊λ̂_{X_{ii},R̃_{i}} X_{ii}

≅ (S , +)^{n−1} ⋊ (S , ∘) . (6.2) Observe also that, for j ≠ i , X_{ij} is normalized by R_{i}, and X_{ii} in particular, so the factors may be placed in any order, yielding, for example, R_{i} = R̃_{i} ∘ X_{ii} = X_{i1} ∘ … ∘ X_{in} (6.3)

(3) Dual formulae and the use of equation (1.2) yield, for each i ,

C_{i} = C̃_{i} + X_{ii} = C̃_{i} ∘ X_{ii} = C_{1i} ∘ … ∘ C_{ni}

≅ C̃_{i} ⋊_{}_{ρ̂\begin{matrix}′ \\ X_{ii},C̃_{i} \end{matrix}} X_{ii}

≅ (S , +)^{n−1} ⋊ (S , ∘) .

(4) For each i < n , T_{i} and R_{i+1} are right ideals of M , T_{i+1} = T_{i} +R_{i+1} and T_{i} ∩R_{i+1} = {0} , so that, by Theorem 5.4, T_{i+1} is the general product T_{i+1} = T_{i} ∘ R_{i+1} ≅ (T_{i} , ∘) ⊛ (R_{i+1} , ∘) . (6.4) (and the general product mappings, though not explicitly described here, may also be deduced from Theorem 5.4). For each i, we have, by a simple induction,

T_{i} = R_{1} ∘ … ∘ R_{i} ≅ (… (R_{1} ⊛ R_{2}) ⊛ …) ⊛ R_{i} .

Steps (1) to (4) culminate, by equation (6.3) and its dual, in the following result. Theorem 6.1. If S is a radical ring and n ≥ 1 then

M_{n}(S) = R_{1} ∘ … ∘ R_{n} = C_{1} ∘ … ∘ C_{n}
= (X_{11} ∘ … ∘ X_{1n}) ∘ … ∘ (X_{n1} ∘ … ∘ X_{nn})
= (X_{11} ∘ … ∘ X_{n1}) ∘ … ∘ (X_{1n} ∘ … ∘ X_{nn})
≅ (… (R_{1} ⊛ R_{2}) ⊛ …) ⊛ R_{n} ≅ (… (C_{1} ⊛ C_{2}) ⊛ …) ⊛ C_{n} .

We describe an alternative recursive decomposition of M = M_{n} , which uses a mixture of general and semidirect products. By equation (6.4) we have the internal general product M = T_{n} = T_{n−1} ∘ R_{n} . (6.5)

But M_{n−1} and C̃_{n} are left ideals of T_{n−1} , M_{n−1} annihilates C̃_{n} by multiplication on the right, T_{n−1} = M_{n−1} + C̃_{n} and M_{n−1} ∩ C̃_{n} = {0} , so, by Corollary 5.3 or 5.5 and the dual of isomorphism (6.1),

T_{n−1} = C̃_{n} ∘ M_{n−1}
≅ (C̃_{n} , ∘) ⋊λ̂_{M_{n−1},C̃_{n}} (M_{n−1} , ∘)
≅ (S , +)^{n−1} ⋊ (M_{n−1} , ∘) .

Thus by equation (6.5) and isomorphism (6.2) we get the following recursive formula.

Theorem 6.2. If S is a radical ring and n ≥ 1 then

(M_{n}(S) , ∘) ≅ ((S , +)^{n−1} ⋊ (M_{n−1}(S) , ∘)) ⊛ ((S , +)^{n−1} ⋊ (S , ∘)) .

7. A group presentation

Let S be any radical ring and n any positive integer. In this section we first find a presentation for ( M_{n}(S) , ∘) in terms of the addition and circle multiplication tables of S . We then modify it to yield a presentation in terms of presentations and sets of normal forms for the groups ( S , + ) and ( S , ∘) . In Section 8 we illustrate how this result can be used to find explicit, concise presentations in important special cases.

Form the alphabet

Σ_{S} = { x_{ij} | x ∈ S , i , j ∈ { 1 , … n } } .

Let R_{S} be the collection of relations of the following types:

(1) ( ∀i , j)( ∀x , y ∈ S)

x_{ij}y_{ij} = \begin{cases}( x + y)_{ij} & if j ≠ i, \\ ( x ∘ y)_{ij} & if j = i . \end{cases}

(2) ( ∀i ≠ l , j ≠ k)( ∀x , y ∈ S) [ x_{ij} , y_{kl}] = 1 .

(3) ( ∀i ≠ j ≠ k ≠ i)( ∀x , y ∈ S)

[ x_{ij} , y_{jk}] = ( −xy)_{ik} .

(4) ( ∀i ≠ j)( ∀x , y ∈ S)

[ x_{ii} , y_{ij}] = ( x′y)_{ij} .

(5) ( ∀i ≠ j)( ∀x , y ∈ S)

[ x_{ij} , y_{jj}] = ( −xy)_{ij} .

(6) ( ∀i > j)( ∀x , y ∈ S)

x_{ij}y_{ji} = ( ( −yx)′_{jj} ) y_{ji}x_{ij}( −xy)_{ii} .

The reader might observe that (3) and (5) could be amalgamated. However it is convenient to keep them separate for the purposes of the proofs below.

Theorem 7.1. (M_{n}(S) , ∘) ≅ ⟨ Σ_{S} | R_{S} ⟩ .

Proof. Put G = ⟨ Σ_{S} | R_{S} ⟩ . We identify elements of G with words over Σ_{S} without causing confusion. Observe that

Σ_{S} = ⋃_{i ,j}X_{ij}

where, for each i , j ,

X_{ij} = { x_{ij} | x ∈ S } ,

so that, by (1) of R_{S}, X_{ij} = ⟨ X_{ij} ⟩ (where the angular brackets may be interpreted both as subgroup and submonoid generation). By rearranging the commutators it is easy to see that the relations in R_{S} imply that the hypotheses of Lemma 3.5 are satisfied with G in place of M, so

G = ∏_{i=1}^{n} ∏_{j=1}^{n} X_{ij} . (7.1)

For each x ∈ S and i , j , let x_{ij}^{†} denote the n × n matrix consisting of zeros everywhere except for x in the ( i , j)th place. It is routine to check that all of the relations of R_{S} become equations in M_{n}(S) when each x_{ij} is replaced by x_{ij}^{†} . As an example of the nature of the calculations involved, the following suffices to verify (6):

j  i
\begin{matrix}j \\ i \end{matrix} (\begin{matrix}−( yx)′ & 0 \\ 0 & 0 \end{matrix}) ∘ (\begin{matrix}0 & y \\ 0 & 0 \end{matrix}) ∘ (\begin{matrix}0 & 0 \\ x & 0 \end{matrix}) ∘ (\begin{matrix}0 & 0 \\ 0 & −xy \end{matrix})

= (\begin{matrix}( −yx)′ & y − ( yx)′y \\ 0 & 0 \end{matrix}) ∘ (\begin{matrix}0 & 0 \\ x & −xy \end{matrix})
= (\begin{matrix}( −yx)′ − ( y − ( yx)′y)x & y − ( yx)′y + ( y − ( yx)′y)xy \\ x & −xy \end{matrix})
= (\begin{matrix}0 & y \\ x & −xy \end{matrix}) = (\begin{matrix}0 & 0 \\ x & 0 \end{matrix}) ∘ (\begin{matrix}0 & y \\ 0 & 0 \end{matrix})

Thus the mapping x_{ij} ↦ x_{ij}^{†} ( x ∈ S , i , j ∈ { 1 , …, n }) induces a well-defined homomorphism φ : G ⟶ M_{n}(S) . Now put, for i , j ,

X_{ij}^{†} = { x_{ij}^{†} | x ∈ S }

and observe, by Theorem 6.1, that

M_{n}(S) = ( X_{11}^{†} ∘ … ∘ X_{1n}^{†}) ∘ … ∘ ( X_{n1}^{†} ∘ … ∘ X_{nn}^{†}) = ⟨ ⋃_{i ,j}X_{ij}^{†} ⟩_{∘} ,

which proves φ is onto. To complete the proof it suffices to show φ is one-one, and for that it is sufficient, by equation (7.1), to check the induced map, also denoted by φ , on the set of words

W = ∏_{i=1}^{n} ∏_{j=1}^{n} X_{ij}

is one-one.

Let u , v ∈ W and suppose uφ = vφ . There are elements x( i , j) , y( i , j) ∈ S for i , j ∈ { 1 , …, n } such that

u = ∏_{i=1}^{n} ∏_{j=1}^{n}( x( i, j) )_{ij} and v = ∏_{i=1}^{n} ∏_{j=1}^{n}( y( i , j) )_{ij} .

For each k ∈ { 1 , …, n } , put

r_{k} = ∏_{j=1}^{n}( x( k , j) )_{kj} and s_{k} = ∏_{j=1}^{k}( y( k , j) )_{kj} ,

so u = r_{1} … r_{n} and v = s_{1} … s_{n} . We will prove

(7.2)  ( ∀k = 1 , …, n) r_{k}φ = s_{k}φ .

Suppose that r_{l}φ = s_{l}φ for all l > k ( which is vacuously true if k = n). Then, since M_{n}(S) is radical,

( r_{1} … r_{k})φ = ( r_{1} … r_{k})φ ∘ ( r_{k+1} … r_{n})φ ∘ ( ( r_{k+1} … r_{n})φ)′
= ( r_{1} … r_{n})φ ∘ ( r_{n}φ)′ ∘ … ∘ ( r_{k+1}φ)′
= ( s_{1} … s_{n})φ ∘ ( s_{n}φ)′ ∘ … ∘ ( s_{k+1}φ)′
= ( s_{1} … s_{k})φ .

But, by a simple matrix calculation, we see that the kth rows of ( r_{1} … r_{k})φ and ( s_{1} … s_{k})φ are r_{k}φ and s_{k}φ respectively, whence r_{k}φ = s_{k}φ , and equation (7.2) follows by induction.

By another simple matrix calculation, r_{k}φ and s_{k}φ are matrices with zeros everywhere except for the kth rows which are

( x( k , 1 ) … x( k , k) ẑx( k , k + 1 ) … ẑx( k , n) )

and

( y( k , 1 ) … y( k , k) ŵ y( k , k + 1 ) … ŵ y( k , n) )

respectively, where z = x( k , k) and w = y( k , k) . But r_{k}φ = s_{k}φ , so, for j = 1 , …, k , x( k , j) = y( k , j) , and in particular z = w . Hence, also, for j = k + 1 , …, n,

x( k , j) = \overline{z} ẑx( k , j) = \overline{z} ẑy( k , j) = y( k , j) .

This proves u = v , proving φ is one-one, completing the proof of Theorem 7.1. □

The reader might note that if S is finite then it is not necessary to argue that ϕ is oneone on W, since this follows immediately from the fact that ϕ is onto and, by Lemma 3.5, |G| ≤ Π_{i=1}^{n}Π_{j=1}^{n} |X_{ij}|.

The presentation of Theorem 7.1 uses the entire addition and circle multiplication tables of S , leading to superfluity in practice. For example, the generators 0_{ij} may be deleted and replaced by 1 throughout, for all i , j . In Theorem 7.2 below we give a presentation for ( M_{n}(S) , ∘) in terms of presentations for ( S , + ) and ( S , ∘) . Suppose

( S , + ) ≅ ⟨ Γ^{(+)} | R^{(+)} ⟩ , ( S , ∘) ≅ ⟨ Γ^{( ∘)} | R^{(∘)} ⟩

for some alphabet Γ^{(+)} , Γ^{(∘)} and collections of relations R^{(+)} , R^{(∘)} over Γ^{(+)} , Γ^{(∘)} respectively. We may suppose no generator is redundant, so that there are collections W^{(+)} , W^{(∘)} of words, which we may refer to as normal forms, over Γ^{(+)} , Γ^{(∘)} respectively such that

Γ^{(+)} ⊆ W^{(+)} , Γ^{(∘)} ⊆ W^{(∘)} ,

and bijections

φ : W^{(+)} ⟶ S , ψ : W^{(∘)} ⟶ S

whose inverses induce the above isomorphisms. We now create a new alphabet

Γ = { σ_{ij} | i , j ∈ { 1 , …, n } , σ ∈ Γ^{(+)} if i ≠ j , and σ ∈ Γ^{(∘)} if i = j } .

For any i , j ∈ { 1 , …, n } , put

1_{ij} = 1

where 1 here denotes the empty word, and if w = σ^{(1)} … σ^{(m)} is any non-empty word where σ^{(1)} , …, σ^{(m)} are letters, put

w_{ij} = σ_{ij}^{(1)} … σ_{ij}^{(m)} ,

so that, if w is over Γ^{(+)} and i ≠ j , or over Γ^{(∘)} and i = j , then w_{ij} is over Γ . For any i ≠ j, let R_{ij}^{(+)} denote the collection of relations of the form

v_{ij} = w_{ij}

where v = w is a relation of R^{(+)} . For any i, let R_{i}^{(∘)} denote the collection of relations of the form

v_{ii} = w_{ii}

where v = w is a relation of R^{(∘)} . Now let R denote the collection of relations of the following types:

(1) ⋃_{i≠j} R_{ij}^{(+)} ∪ ⋃_{i} R_{i}^{(∘)} .

(2) ( ∀i ≠ l , j ≠ k) ( ∀a ∈ \begin{cases}Γ^{(+)} & if i ≠ j \\ Γ^{(∘)} & if i = j \end{cases} ) ( ∀b ∈ \begin{cases}Γ^{(+)} & if k ≠ l \\ Γ^{(∘)} & if k = l \end{cases} )

[ a_{ij} , b_{kl}] = 1 .

(3) ( ∀i ≠ j ≠ k ≠ i)( ∀u , v ∈ W^{(+)} )

[ u_{ij} , v_{jk}] = ( ( −( uφ)( vφ) )φ^{−1} )_{ik} .

(4) ( ∀i ≠ j)( ∀u ∈ W^{(∘)} , v ∈ W^{(+)} )

[ u_{ii} , v_{ij}] = ( ( ( uψ)′( vφ) )φ^{−1} )_{ij} .

(5) ( ∀i ≠ j)( ∀u ∈ W^{(+ )} , v ∈ W^{(∘)} )

[ u_{ij} , v_{jj}] = ( ( −( uφ)( vψ) )φ^{−1} )_{ij} .

(6) ( ∀i > j)( ∀u , v ∈ W^{(+)} )

u_{ij}v_{ji} = ( ( −( vφ)( uφ) )′ψ^{−1} )_{jj} v_{ji}u_{ij} ( ( −( uφ)( vφ) )ψ^{−1} )_{ii} .

Theorem 7.2. ( M_{n}(S) , ∘) ≅ ⟨ Γ | R ⟩ .

Proof. Put H = ⟨ Γ | R ⟩ . By Theorem 7.1 it is sufficient to prove H ≅ G = ⟨ Σ_{S} | R_{S} ⟩ . We do this by applying Tietze transformations to G . Step 1: Add to Σ_{S} the alphabet Γ , which we may assume to be disjoint from Σ_{S} , and to R_{S}

relations of the form

σ_{ij} = s_{ij}

where σ ∈ \begin{cases}Γ^{(+)} & if i ≠ j \\ Γ^{(∘)} & if i = j \end{cases} and s = \begin{cases}σφ & if i ≠ j \\ σψ & if i = j . \end{cases}

Step 2: Add relations

x_{ij} = w_{ij}

where x ∈ S and w = \begin{cases}xφ^{−1} & if i ≠ j \\ xψ^{−1} & if i = j . \end{cases} This is justified for i ≠ j ( i = j being

similar ) as follows:

Suppose x ∈ S and xφ^{−1} = w = σ^{(1)} … σ^{(t)} for some σ^{(1)} , …, σ^{(t)} ∈ Γ^{(+)} . Then, in S,

x = σ^{(1)}φ + … + σ^{(t)}φ

so, by type (1) relations in G , followed by Step 1 relations,

x_{ij} = ( σ^{(1)}φ)_{ij} … ( σ^{(t)}φ)_{ij} = σ^{(1)}_{ij} … σ^{(t)}_{ij} = w_{ij} .

Step 3: Add all relations in

⋃_{i≠j} R_{ij}^{(+)} ∪ ⋃_{i} R_{i}^{(∘)} .

This is justified for R^{(+)} ( R^{(∘)} is similar ) as follows: Suppose v = w is a relation of R^{(+)} . Write v = σ^{(1)} … σ^{(t)} , w = τ^{(1)} … τ^{(u)} where σ^{(1)} , …, σ^{(t)} , τ^{(1)} , …, τ^{(u)} ∈ Γ^{(+)} . Then, in S ,

σ^{(1)}φ + … + σ^{(t)}φ = τ^{(1)}φ + … + τ^{(u)}φ ,

so, in G , using type (1) relations,

( σ^{(1)}φ)_{ij} … ( σ^{(t)}φ)_{ij} = ( σ^{(1)}φ + … + σ^{(t)}φ)_{ij}
= ( τ^{(1)}φ + … + τ^{(u)}φ)_{ij}
= ( τ^{(1)}φ)_{ij} … ( τ^{(u)}φ)_{ij} ,

so, by Step 1 relations, we deduce that

v_{ij} = σ_{ij}^{(1)} … σ_{ij}^{(t)} = (σ^{(1)}φ)_{ij} … (σ^{(t)}φ)_{ij}
= (τ^{(1)}φ)_{ij} … (τ^{(t)}φ)_{ij} = τ_{ij}^{(1)} … τ_{ij}^{(u)} = w_{ij} .

Step 4: Remove all of G’s type (1) relations. This is justified for i ≠ j ( i = j being similar ) as follows: Suppose x , y ∈ S . Then x + y ∈ S and, in ⟨ Γ^{(+)} | R^{+} ⟩ ,

( xφ^{−1})( yφ^{−1} ) = ( x + y)φ^{−1} ,

so, as a consequence of relations in R_{ij}^{(+)} , we have

( xφ^{−1} )_{ij}( yφ^{−1} )_{ij} = (( x + y)φ^{−1})_{ij} ,

from which we derive, using relations from Step 2, the relation

x_{ij}y_{ij} = ( x + y)_{ij} .

Step 5: Remove all letters from Σ_{S} , replacing any letter s_{ij} ( s ∈ S , i , j ∈ { 1 , …, n }) which appears in any relation by the word ( sφ^{−1} )_{ij} if i ≠ j , and ( sψ^{−1} )_{ij} if i = j . The Step 1, 2 relations now become equality of the same letter, word respectively, so are redundant and may be deleted.

The effect of Steps 1 to 5 is to replace Σ_{S} by Γ , type (1) relations of R_{S} by type (1) relations of R and each relation of types (2) to (6) of R_{S} by a relation in which any letter s_{ij} has been replaced by the word ( sφ^{−1} )_{ij} or ( sψ^{−1} )_{ij} if i ≠ j or i = j respectively.

Step 6: Consider i ≠ l , j ≠ k . The transformed type (2) relations have the form

[ v_{ij} , w_{kl}] = 1

where v ∈ \begin{cases}W^{(+)} & if i ≠ j \\ W^{(∘)} & if i = j \end{cases} , w ∈ \begin{cases}W^{(+)} & if k ≠ l \\ W^{(∘)} & if k = l , \end{cases} .

Delete all such relations where v ∉ \begin{cases}Γ^{(+)} & if i ≠ j \\ Γ^{(∘)} & if i = j \end{cases}

or w ∉ \begin{cases}Γ^{(+)} & if k ≠ l \\ Γ^{(∘)} & if k = l \end{cases} , since these are implied by the remaining relations of the

form [ σ_{ij} , τ_{kl}] = 1 where σ ∈ \begin{cases}Γ^{(+)} & if i ≠ j \\ Γ^{(∘)} & if i = j \end{cases} and τ ∈ \begin{cases}Γ^{(+)} & if k ≠ l \\ Γ^{(∘)} & if k = l . \end{cases}

The relations that remain are precisely those catalogued by R , and Theorem 7.2 is proved. □

8. Examples

We apply the results of the previous section section to determine ( M_{n}( S) , ∘) when S = pℤ_{p^{t}} , the radical of the ring ℤ_{p^{t}} , where p is a prime and t ≥ 3 ( the cases t = 1 , 2 yielding trivial, elementary abelian groups respectively ). The cases p odd and even are treated separately because ( pℤ_{p^{t}} , ∘) is cyclic, in fact isomorphic to ( pℤ_{p^{t}} , + ) , when p is odd, but noncyclic when p = 2 .

Theorem 8.1. Let p be odd. Then

( M_{n}( pℤ_{p^{t}}) , ∘) ≅ ⟨ Γ | R ⟩

over the alphabet

Γ = { a_{ij} | i , j ∈ { 1 , …, n } }

and where R comprises relations of the following types ( 1 ) ( ∀i , j) a_{ij}^{p^{t−1}} = 1 ; ( 2 ) ( ∀i ≠ l , j ≠ k) [ a_{ij} , a_{kl}] = 1 ; ( 3 ) ( ∀i ≠ j ≠ k ≠ i) [ a_{ij} , a_{jk}] = a_{ik}^{−p};

( 4 ) ( ∀i ≠ j) [ a_{ii} , a_{ij}] = a_{ij}^{p′}; ( 5 ) ( ∀i ≠ j) [ a_{ij} , a_{jj}] = a_{ij}^{−p}; ( 6 ) ( ∀i > j)( ∀m = 0 , …, p^{t−3} − 1 )

a_{ij}^{1−( −mp^{2} )′}a_{ji} = a_{jj}^{−α}a_{ji}a_{ij}^{1−( −mp^{2} )′}a_{ii}^{α}

where α is the least positive integer such that

( 1 − p)^{α} = 1 + ( 1 − ( −mp^{2} )′)p^{2}

in ℤ_{p^{t}} ( which exists because ( pℤ_{p^{t}} , ∘) is cyclic generated by p).

Proof. Observe that, for S = pℤ_{p^{t}} ,

( S , + ) = ⟨ p ⟩_{+} ≅ ⟨ a | a^{p^{t−1}} = 1 ⟩ ≅ ⟨ p ⟩_{∘} = ( S , ∘) .

In the framework leading up to Theorem 7.2 we take

W^{(+)} = W^{(∘)} = { a^{m} | m ∈ { 0 , …, p^{t−1} − 1 } }

and choose bijections

φ : W^{(+)} ⟶ ( S , + ) , a^{m} ↦ mp

and

ψ : W^{(∘)} ⟶ ( S , ∘) , a^{m} ↦ p^{∘m} .

Then relations of type ( 1 ), ( 2 ) are identical in Theorems 7.2 and 8.1.

Consider relations of type ( 3 ) in Theorem 7.2. Suppose i ≠ j ≠ k ≠ i . The relations have the form

[ a_{ij}^{λ} , a_{jk}^{μ}] = a_{ik}^{−λμp}

for λ , μ ∈ { 0 , …, p^{t−1} } . The cases λ = 0 or μ = 0 are redundant, and the cases λ > 0 and μ > 0 follow from the relation [ a_{ij} , a_{jk}] = a_{ik}^{−p}, by Lemma 3.1, so may be deleted, leaving type ( 3 ) relations of Theorem 8.1.

Suppose i ≠ j . Relations of type ( 4 ) of Theorem 7.2 have the form

[ a_{ii}^{λ} , a_{ij}^{μ}] = ( ( ( p^{∘λ})′( μp) )φ^{−1} )_{ij} = a_{ij}^{( 1−( 1−p′)^{λ})μ}

and of type ( 5 ) the form

[ a_{ij}^{λ} , a_{jj}^{μ}] = ( ( −( λp)( p^{∘μ}) )φ^{−1} )_{ij} = a_{ij}^{−( 1−( 1−p)^{μ})λ} .

The cases λ = 0 or μ = 0 are redundant, and the cases λ > 0 and μ > 0 follow from the relations

[ a_{ii} , a_{ij}] = a_{ij}^{p′} and [ a_{ij} , a_{jj}] = a_{ij}^{−p} ,

by Lemma 3.2, noting that the latter is equivalent to

[ a_{jj} , a_{ij}] = [ a_{ij} , a_{jj}]^{−1} = a_{ij}^{p} ,

whilst the type ( 5 ) relation is equivalent to

[ a_{jj}^{μ} , a_{ij}^{λ}] = a_{ij}^{( 1−( 1−p)^{μ})λ} .

Deleting all but the cases λ = μ = 1 yields type ( 4 ) and ( 5 ) relations of Theorem 8.1.

Suppose i > j . Relations of type ( 6 ) of Theorem 7.2 have the form

a_{ij}^{λ}a_{ji}^{μ} = ( ( −( μp)( λp) )′ψ^{−1} )_{jj}a_{ji}^{μ}a_{ij}^{λ}( ( −( λp)( μp) )ψ^{−1} )_{ii}
= a_{jj}^{−ν}a_{ji}^{μ}a_{ij}^{λ}a_{ii}^{ν}

where ν is the least nonnegative integer such that

( 1 − p)^{ν} = 1 + λμp^{2} ,

for λ , μ ≥ 0 . We delete all such relations except for the cases μ = 1, λ = 1 − ( −mp^{2} )′ for m = 0 , …, p^{t−3} − 1 , since those to be deleted are either redundant ( when λ = 0 or μ = 0 ), or follow from the ones to be retained and the relations [ a_{jj} , a_{ii}] = 1 (type 2) and

a_{ij}^{a_{jj}} = a_{ij}^{1−p} , a_{ij}^{a_{ii}} = a_{ij}^{1−p′} , a_{ji}^{a_{jj}} = a_{ji}^{1−p′} , a_{ji}^{a_{ii}} = a_{ji}^{1−p}

( the result of rearranging type ( 4 ) and ( 5 ) relations of Theorem 8.1 ) by Lemma 3.4. This completes the proof of Theorem 8.1. □

This presentation simplifies further when t = 3 .

Theorem 8.2. Let p be odd. Then

( M_{n}( pℤ_{p^{3}} ) , ∘) ≅ ⟨ Γ | R ⟩

over the alphabet

Γ = { a_{ij} | i , j ∈ { 1 , …, n } }

where R consists of relations ( 1 ) ( ∀i , j) a_{ij}^{p^{2}} = 1 ; ( 2 ) ( ∀i ≠ l , j ≠ k) [ a_{ij} , a_{kl}] = 1 ; ( 3 ) ( ∀i , j , k , k ≠ i) [ a_{ij} , a_{jk}] = a_{ik}^{−p}; ( 4 ) ( ∀i > j) [ a_{ij} , a_{ji}] = a_{jj}^{p}a_{ii}^{−p} .

Proof. Relations ( 1 ), ( 2 ) are the same as those of Theorem 8.1. Note that in ℤ_{p^{3}} , p′ = −p − p^{2} , so that relations ( 4 ) of Theorem 8.1 become

[ a_{ii} , a_{ij}] = a_{ij}^{−p−p^{2}} = a_{ij}^{−p} ,

by ( 1 ). This can be amalgamated with ( 3 ) and ( 5 ) of Theorem 8.1 to yield ( 3 ) of Theorem 8.2. There is only one relation in ( 6 ) of Theorem 8.1 when t = 3 :

a_{ij}a_{ji} = a_{jj}^{−( p^{2}−p)}a_{ji}a_{ii}a_{ii}^{p^{2}−p}

since ( 1 − p)^{p^{2}−p} = 1 + p^{2} in ℤ_{p^{3}} , which becomes, by ( 1 ),

a_{ij}a_{ji} = a_{jj}^{p}a_{ji}a_{ii}a_{ii}^{−p} .

Tracing through the isomorphism with M_{n}( ℤ_{p^{3}} ) ( induced by the map a_{kl} ↦ a_{kl}^{†} in the notation of Theorem 7.1 ) we see that a_{jj}^{p} and a_{ii}^{−p} are central ( since they correspond with matrices with one nonzero entry equal to p^{2} and −p^{2} respectively ), so the above relation, in conjunction with commutativity relations, is equivalent to

[ a_{ij} , a_{ji}] = a_{jj}^{p}a_{ii}^{−p} . □

Example 8.3. Further simplification takes place when n = 2 in Theorem 8.2: one relation suffices in each of ( 2 ) and ( 4 ), and relations of type ( 3 ) may be rearranged to give four conjugation relations, yielding the following presentation for ( M_{2}( pℤ_{p^{3}} ) , ∘) , for any odd prime p :

⟨ a_{11} , a_{12} , a_{21} , a_{22} | a_{ij}^{p^{2}} = 1 ( ∀i , j) , [ a_{11} , a_{22} ] = 1 , [ a_{21} , a_{12} ] = a_{11}^{p}a_{22}^{−p} ,
a_{ij}^{a_{ii}} = a_{ij}^{1+p} , a_{ij}^{a_{jj}} = a_{ij}^{1−p} ( ∀i ≠ j) ⟩

which the reader will recognize as being the presentation of the general product G ⊛ G of Example 2.2 , after renaming generators.

Finally we consider the case when p = 2 .

Theorem 8.4.

( M_{n}( 2ℤ_{2^{t}}) , ∘) ≅ ⟨ Γ | R ⟩

over the alphabet

Γ = { a_{ij} | i , j ∈ { 1 , …, n } , i ≠ j } ∪ { b_{i} , c_{i} | i ∈ { 1 , …, n } }

where R comprises relations

( ∀i ≠ j) a_{ij}^{2^{t−1}} = 1 ; ( 1 )

( ∀i) b_{i}^{2^{t−2}} = c_{i}^{2} = 1 , [ b_{i} , c_{i}] = 1 ; ( 1 )′

( ∀l ≠ i ≠ j ≠ k ≠ l) [ a_{ij} , a_{kl}] = 1 ; ( 2 )

( 2 )′  ( ∀i ≠ j) [ b_{i} , b_{j}] = [ b_{i} , c_{j}] = [ c_{i} , c_{j}] = 1 ;
( 2 )′′  ( ∀k ≠ i ≠ l ≠ k) [ b_{i} , a_{kl}] = [ c_{i} , a_{kl}] = 1 ;

( ∀i ≠ j ≠ k ≠ i) [ a_{ij} , a_{jk}] = a_{ik}^{−2} ; ( 3 )

( ∀i ≠ j) [ b_{i} , a_{ij}] = a_{ij}^{4′} , [ c_{i} , a_{ij}] = a_{ij}^{2} ; ( 4 )

( ∀i ≠ j) [ a_{ij} , b_{j}] = a_{ij}^{−4} , [ a_{ij} , c_{j}] = a_{ij}^{−2} ; ( 5 )

( ∀i > j)( ∀m = 0 , …, 2^{t−3} − 1 ) ( 6 )

a_{ij}^{1−( −4m)′}a_{ji} = b_{j}^{−α}a_{ji}a_{ij}^{1−( −4m)′}b_{i}^{α}

where α is the least positive integer such that

( −3 )^{α} = 1 + 4( 1 − ( −4m)′)

in ℤ_{2^{t}} ( which exists because all multiples of 4 comprise the

cyclic subgroup of ( 2ℤ_{2^{t}} , ∘) generated by 4 ).

Proof. Observe that, for S = 2ℤ_{2^{t}} ,

( S , + ) = ⟨ 2 ⟩_{+} ≅ ⟨ a | a^{2^{t−1}} = 1 ⟩ ;
( S , ∘) = ⟨ 2 , 4 ⟩_{∘} ≅ ⟨ b , c | b^{2^{t−2}} = c^{2} = 1 , [ b , c] = 1 ⟩ .

In the framework leading up to Theorem 7.2 we take

W^{( + )} = { a^{λ} | λ ∈ { 0 , …, 2^{t−1} − 1 } } ,
W^{( ∘)} = { b^{λ}c^{μ} | λ ∈ { 0 , …, 2^{t−2} − 1 } , μ ∈ { 0 , 1 } } ,
φ : W^{( + )} ⟶ S , a^{λ} ↦ 2λ ,
ψ : W^{( ∘)} ⟶ S , b^{λ}c^{μ} ↦ 2^{∘μ} ∘ 4^{∘λ} .

We apply Theorem 7.2 and also, to decongest notation slightly, identify b_{ii} , c_{ii} with new letters b_{i} , c_{i} respectively. Then relations of types ( 1 ), ( 1 )′, ( 2 ), ( 2 )′ and ( 2 )′′ here are identical with relations of types ( 1 ) and ( 2 ) which feature in Theorem 7.2. The reduction of relations of types ( 3 ) and ( 6 ) is identical to that in the proof of Theorem 8.1, relying on Lemma 3.1 and Lemma 3.4 (with q = 4) respectively.

Suppose i ≠ j. Relations of type (4) of Theorem 7.2 have the form

[b_{i}^{λ}, a_{ij}^{μ}] = (((4^{∘λ})′(2μ))ϕ^{−1})_{ij} = a_{ij}^{(1−(1−4′)^{λ})μ}

or

[b_{i}^{λ}c_{i}, a_{ij}^{μ}] = (((2 ∘ 4^{∘λ})′(2μ))ϕ^{−1})_{ij} = a_{ij}^{(1+(1−4′)^{λ})μ} ,

and of type (5) the form

[a_{ij}^{λ}, b_{j}^{μ}] = ((−(2λ)(4^{∘μ})ϕ^{−1})_{ij} = a_{ij}^{−(1−(−3)^{μ})λ}

or

[a_{ij}^{λ}, b_{j}^{μ}c_{j}] = ((−(2λ)(2 ∘ 4^{∘μ})ϕ^{−1})_{ij} = a_{ij}^{−(1+(−3)^{μ})λ} .

The nonredundant cases follow from the relations

[b_{i}, a_{ij}] = a_{ij}^{4′} , [c_{i}, a_{ij}] = a_{ij}^{2} , [b_{i}, c_{i}] = 1 ,
[a_{ij}, b_{j}] = a_{ij}^{−4} , [a_{ij}, c_{j}] = a_{ij}^{−2} , [b_{j}, c_{j}] = 1 ,

by Lemmas 3.2 and 3.3. Deleting all but the cases λ = μ = 1 yields type (4) and (5) relations of Theorem 8.4, and the proof is complete. □

Again, the presentation simplifies when t = 3 .

Theorem 8.5.

( M_{n}( 2ℤ_{8} ) , ∘) ≅ ⟨ Γ | R ⟩

over the alphabet

Γ = { a_{ij} | i , j ∈ { 1 , …, n } , i ≠ j } ∪ { b_{i} , c_{i} | i ∈ { 1 , …, n } }

where R comprises relations ( 1 ) ( ∀i , j ≠ i) a_{ij}^{4} = b_{i}^{2} = c_{i}^{2} = 1 ; ( 2 ) ( ∀l ≠ i ≠ j ≠ k ≠ l) [ a_{ij} , a_{kl}] = 1 ; ( 3 ) ( ∀i , j ≠ k ≠ l ≠ j) [ b_{i} , c_{j}] = [ b_{k} , b_{l}] = [ b_{i} , a_{kl}]

= [ c_{k} , c_{l}] = [ c_{j} , a_{kl}] = 1 ;

( 4 ) ( ∀i ≠ j ≠ k ≠ i) [ a_{ij} , a_{jk}] = [ c_{i} , a_{ik}] = [ a_{ik} , c_{k}] = a_{ik}^{2} ; ( 5 ) ( ∀i > j) [ a_{ij} , a_{ji}] = b_{j}b_{i} .

Proof. This follows by making economies to relations appearing in Theorem 8.4, noting that 4′ = 4 in ℤ_{8} , exploiting relation ( 1 ), and rearranging ( 6 ) into a commutator relation, noting that b_{i} commutes with a_{ij} and a_{ji} for i ≠ j by ( 4 ) and ( 5 ) . □

Example 8.6. Yet further simplification takes place when n = 2 in Theorem 8.5, yielding the following presentation for ( M_{2}( 2ℤ_{8} ) , ∘) :

⟨ a_{12} , a_{21} , b_{1} , b_{2} , c_{1} , c_{2} | a_{12}^{4} = a_{21}^{4} = b_{i}^{2} = c_{i}^{2} = 1 , a_{12}^{c_{i}} = a_{12}^{−1} , a_{21}^{c_{i}} = a_{21}^{−1} ,
[ b_{i} , x] = [ c_{1} , c_{2} ] = 1 , [ a_{12} , a_{21} ] = b_{1}b_{2} ( ∀i)( ∀x ≠ b_{i}) ⟩

which the reader will recognize as the presentation of the general product H ⊛ H of Example 2.3, after renaming generators.

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Received August 11, 2000