Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 399-406.

Matrices over Centrally ℤ_{2}-graded Rings

Sudarshan Sehgal^{1} Jenő Szigeti^{2}

Department of Mathematics, University of Alberta Edmonton, T6G 2G1, Canada e-mail: s.sehgal@ualberta.ca

Institute of Mathematics, University of Miskolc Miskolc, Hungary 3515 e-mail: matszj@gold.uni-miskolc.hu

Abstract. We introduce a new computational technique for n × n matrices, over a ℤ_{2}-graded ring R = R_{0} ⊕ R_{1} with R_{0} ⊆ Z(R), leading us to a new concept of determinant, which can be used to derive an invariant Cayley-Hamilton identity. An explicit construction of the inverse matrix A^{−1} for any invertible n × n matrix A over a Grassmann algebra E is also obtained. MSC 2000: 16A38, 15A15 (primary); 15A33 (secondary) Keywords: ℤ_{2}-graded ring, skew polynomial ring, determinant and adjoint

1. Introduction

The main aim of the present paper is to introduce a new computational technique for matrices over certain ℤ_{2}-graded rings. We shall consider n × n matrices over a ℤ_{2}-graded ring R = R_{0} ⊕ R_{1} with the property R_{0} ⊆ Z(R), where Z(R) denotes the centre of R. For these matrices our method provides a possibility to use the classical determinant theory of matrices over commutative rings. The most important example for ℤ_{2}-graded rings with the above mentioned property is the exterior (Grassmann) algebra

E = F ⟨v_{1}, v_{2}, …, v_{i}, … | v_{i}v_{j} = −v_{j}v_{i} for all integers 1 ≤ i ≤ j⟩

1Research supported by NSERC grant A-5300 2The second author was supported by OTKA of Hungary, grant no. T 029525 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

400 S. Sehgal, J. Szigeti: Matrices over Centrally ℤ_{2}-graded Rings

and the polynomial algebra E[t], where F is a field, t is a commuting indeterminant. The ℤ_{2}- gradings are E = E_{0} ⊕ E_{1} and E[t] = E_{0}[t] ⊕ E_{1}[t] with E_{0} being the subspace (subalgebra) generated by the monomials of even length and E_{1} being the subspace generated by the monomials of odd length. We note that the F -algebra M_{n}(E) of n × n matrices over the (infinite dimensional) exterior algebra E with char(F) = 0 is one of the most important objects of study in the theory of PI-algebras (see Kemer’s structure theory of T-ideals in [2] and [3]).

First of all, our technique leads to a new concept of invariant determinant, which can be used to derive an invariant Cayley-Hamilton identity in M_{n}(R). An immediate application of our results will provide a new explicit construction of the inverse matrix A^{−1} for any invertible n × n matrix A ∈ M_{n}(E).

Since the existence of a ℤ_{2}-grading R = R_{0} ⊕ R_{1} with the property R_{0} ⊆ Z(R) implies that R is Lie nilpotent of index 2, it would be desirable to find the precise relationship between the concepts presented in the sequel and the Lie nilpotent determinant theory in [4]. The constructions in [4] are based on the use of the so called preadjoint, which is a natural but complicated generalization of the ordinary adjoint matrix. In defining our determinant, here we use only classical determinants and adjoints. Our results on n × n matrices over R = R_{0} ⊕ R_{1} with R_{0} ⊆ Z(R) are similar to the results of [4] specialized to n × n matrices over Lie nilpotent rings of index 2. We believe that our present approach is easier to understand and gives more chance to find an explicit form of the Cayley-Hamilton equation (Newton formulae) for n ≥ 3. Using sophisticated calculations, starting from the characteristic polynomial defined in [4], M. Domokos obtained Newton formulae for 2 × 2 matrices over the Grassmann algebra (see [1]).

2. ℤ_{2}-gradings and skew polynomial rings

A ℤ_{2}-grading of an (associative) ring R is a pair (R_{0}, R_{1}), where R_{0} and R_{1} are additive subgroups of R such that R = R_{0} ⊕ R_{1} and R_{i}R_{j} ⊆ R_{i+j} for all i, j ∈ {0, 1} and i + j is taken modulo 2. The relation R_{0}R_{0} ⊆ R_{0} ensures that R_{0} is a subring of R. Now any element r ∈ R can be uniquely written as r = r_{0} + r_{1}, where r_{0} ∈ R_{0} and r_{1} ∈ R_{1}. It is easy to see that the existence of 1 ∈ R implies that 1 ∈ R_{0}. The function σ : R ⟶ R defined by σ(r_{0} + r_{1}) = r_{0} − r_{1} is a ring homomorphism (actually, it is an automorphism of R). A more general situation is, when R is considered as a C-algebra for some commutative ring C ⊆ Z(R) and R = ⊕_{u∈S}R_{u} is graded by a subsemigroup S ⊆ U(C) of the multiplicative group

of units in C (each R_{u} ⊆ R is a C-submodule) and σ( ∑_{u∈S}r_{u}) = ∑_{u∈S}ur_{u}.

For a ℤ_{2}-graded ring R = R_{0} ⊕ R_{1} let us consider the skew polynomial ring R[x, σ] in the skew indeterminate x. The elements of R[x, σ] are left polynomials of the form f(x) = a_{0} + a_{1}x + ⋯ + a_{k}x^{k} with a_{0}, a_{1}, …, a_{k} ∈ R. Besides the obvious addition, we have the following multiplication rule in R[x, σ]:

xr = σ(r)x for all r ∈ R, i.e. that x(r_{0} + r_{1}) = (r_{0} − r_{1})x for all r_{0} ∈ R_{0}, r_{1} ∈ R_{1}

and

(a_{0} + a_{1}x + ⋯ + a_{k}x^{k})(b_{0} + b_{1}x + ⋯ + b_{l}x^{l}) = c_{0} + c_{1}x + ⋯ + c_{k+l}x^{k+l},

where

c_{m} = ∑_{i+j=m,i≥0,j≥0}a_{i}σ^{i}(b_{j}).

Since σ is an involution, x^{2} is a central element of R[x, σ]: we have σ(σ(r)) = r and x^{2}r = xσ(r)x = σ(σ(r))x^{2} = rx^{2} for all r ∈ R, moreover x^{2} commutes with the powers of x. Thus the ideal (x^{2}) ◁ R[x, σ] generated by x^{2} can be written as (x^{2}) = R[x, σ]x^{2} = x^{2}R[x, σ]. Consider the factor ring R[x, σ]/(x^{2}), then for any element f(x) ∈ R[x, σ] there exists exactly one left polynomial of the form r + sx ∈ R[x, σ] in the residue class f(x) + (x^{2}). Hence the elements of R[x, σ]/(x^{2}) can be represented by linear left polynomials with coefficients in R and the multiplication in R[x, σ]/(x^{2}) is the following:

(r + sx)(p + qx) = rp + (rq + sσ(p))x,

where r, s, p, q ∈ R. The above observation ensures that R[x, σ]/(x^{2}) = R ⊕ Rx is a ℤ_{2}- grading with (Rx)(Rx) = {0}. It follows that the n × n matrices P, Q ∈ M_{n}(R[x, σ]/(x^{2})) can be uniquely written as P = P′ +P′′x and Q = Q′ +Q′′x for some P′, P′′, Q′, Q′′ ∈ M_{n}(R) and that

(∗)  P Q = P′Q′ + (P′Q′′ + P′′σ(Q′))x,

where σ(Q′) = [σ(q′_{ij})] is the natural action of σ on Q′ = [q′_{ij}] and the products P′Q′, P′Q′′, P′′σ(Q′) are taken in M_{n}(R). It can be easily seen, that

\overline{R} = \{r_{0} + s_{1}x | r_{0} ∈ R_{0} and s_{1} ∈ R_{1}\} ⊆ \{r + sx | r, s ∈ R\} = R[x, σ]/(x^{2})

is a subring of R[x, σ]/(x^{2}). Indeed, (r_{0} + s_{1}x)(p_{0} + q_{1}x) = r_{0}p_{0} + (r_{0}q_{1} + s_{1}σ(p_{0}))x, where r_{0}p_{0} ∈ R_{0} and r_{0}q_{1} + s_{1}σ(p_{0}) = r_{0}q_{1} + s_{1}p_{0} ∈ R_{1} for all r_{0}, p_{0} ∈ R_{0} and s_{1}, q_{1} ∈ R_{1}. In consequence, \overline{R} = R_{0} ⊕ R_{1}x is a ℤ_{2}-grading. We note that \overline{R} can be defined directly on the product R_{0} ×R_{1} with componentwise addition and taking the multiplication (r_{0}, s_{1})(p_{0}, q_{1}) = (r_{0}p_{0}, r_{0}q_{1} + s_{1}p_{0}). If R_{0} ⊆ Z(R) with Z(R) being the centre of R, then \overline{R} is commutative:

(r_{0} + s_{1}x)(p_{0} + q_{1}x) = r_{0}p_{0} + (r_{0}q_{1} + s_{1}p_{0})x =
= p_{0}r_{0} + (p_{0}s_{1} + q_{1}r_{0})x = (p_{0} + q_{1}x)(r_{0} + s_{1}x).

The condition R_{0} ⊆ Z(R) also implies the Lie nilpotence (of index 2) of R. For the elements r, s ∈ R we have r = r_{0}+r_{1}, s = s_{0}+s_{1} for some r_{0}, s_{0} ∈ R_{0} and r_{1}, s_{1} ∈ R_{1}. Now r_{0}, s_{0} ∈ Z(R) implies that [r_{0}, s] = [r_{1}, s_{0}] = 0, whence we get [r, s] = [r_{0} + r_{1}, s] = [r_{0}, s] + [r_{1}, s] = [r_{1}, s_{0} + s_{1}] = [r_{1}, s_{0}] + [r_{1}, s_{1}] = [r_{1}, s_{1}] = r_{1}s_{1} − s_{1}r_{1} ∈ R_{0}. Thus [r, s] ∈ Z(R), so we obtain that [[r, s], w] = 0 for all r, s, w ∈ R.

3. Computing with n × n matrices over a centrally ℤ_{2}-graded ring

A ℤ_{2}-grading (R_{0}, R_{1}) of the ring R is called central, if R_{0} ⊆ Z(R). Let A = [a_{ij}] ∈ M_{n}(R) be an n × n matrix over a ring with a central ℤ_{2}-grading, then a_{ij} = a_{ij}^{(0)} + a_{ij}^{(1)} for some unique a_{ij}^{(0)} ∈ R_{0} and a_{ij}^{(1)} ∈ R_{1} for all integers 1 ≤ i ≤ n and 1 ≤ j ≤ n, i.e. A = A_{0} + A_{1} with A_{0} = [a_{ij}^{(0)}] ∈ M_{n}(R_{0}) and A_{1} = [a_{ij}^{(1)}] ∈ M_{n}(R_{1}). The companion matrix of A in M_{n}(\overline{R}) is defined as

A_{0} + A_{1}x = [a_{ij}^{(0)} + a_{ij}^{(1)}x].

Since \overline{R} is commutative, the determinant and the adjoint of A_{0} + A_{1}x are defined and can be written as

det(A_{0} + A_{1}x) = d_{0} + d_{1}x ∈ \overline{R}

and

adj(A_{0} + A_{1}x) = [b_{ij}^{(0)} + b_{ij}^{(1)}x] = B_{0} + B_{1}x ∈ M_{n}(\overline{R}),

where d_{0} ∈ R_{0} and d_{1} ∈ R_{1} are elements, B_{0} = [b_{ij}^{(0)}] ∈ M_{n}(R_{0}) and B_{1} = [b_{ij}^{(1)}] ∈ M_{n}(R_{1}) are n × n matrices, each of these objects is uniquely determined by A. Clearly, d_{0} =det(A_{0}), B_{0} =adj(A_{0}) and the elements d_{1}, b_{ij}^{(1)} ∈ R_{1} are also polynomial expressions of the a_{ij}^{(0)}’s and the a_{ij}^{(1)}’s (it is not hard to give them explicitly).

3.1. Theorem. The elements of the product matrices

A(B_{0} + B_{1}) = (A_{0} + A_{1})(B_{0} + B_{1}) and (B_{0} + B_{1})A = (B_{0} + B_{1})(A_{0} + A_{1})

are contained in the subring R_{0}[d_{1}] of R generated by d_{1} and the elements of R_{0}, namely:

A(B_{0} + B_{1}), (B_{0} + B_{1})A ∈ M_{n}(R_{0}[d_{1}]).

Proof. Since d_{0} + d_{1}x is the determinant and B_{0} + B_{1}x is the adjoint of A_{0} + A_{1}x, we have

(A_{0} + A_{1}x)(B_{0} + B_{1}x) = (d_{0} + d_{1}x)I,

in M_{n}(\overline{R}), where I is the identity matrix. In view of

M_{n}(\overline{R}) = M_{n}(R_{0} ⊕ R_{1}x) ⊆ M_{n}(R ⊕ Rx) = M_{n}(R[x, σ]/(x^{2}))

and σ(B_{0}) = B_{0}, the application of (∗) gives that

A_{0}B_{0} + (A_{0}B_{1} + A_{1}B_{0})x = d_{0}I + (d_{1}I)x,

where A_{0}B_{0} and A_{0}B_{1} + A_{1}B_{0} are taken in M_{n}(R). Using the unique r_{0} + s_{1}x form (with r_{0} ∈ R_{0} and s_{1} ∈ R_{1}) of the elements in \overline{R} and matching the coefficients of x in the left and the right side of the above equation, we obtain the following identity in M_{n}(R):

A_{0}B_{1} + A_{1}B_{0} = d_{1}I.

Thus

A(B_{0} + B_{1}) = (A_{0} + A_{1})(B_{0} + B_{1}) = (A_{0}B_{0} + A_{1}B_{1}) + (A_{0}B_{1} + A_{1}B_{0}) =
(A_{0}B_{0} + A_{1}B_{1}) + d_{1}

and A_{0}B_{0} + A_{1}B_{1} ∈ M_{n}(R_{0}) imply that A(B_{0} + B_{1}) ∈ M_{n}(R_{0}[d_{1}]). The similar statement on the product (B_{0} + B_{1})A can be proved analogously. □

The condition R_{0} ⊆ Z(R) implies that the subring R_{0}[d_{1}] ⊆ R is commutative (the elements of R_{0}[d_{1}] are polynomials of d_{1} with coefficients in R_{0}). As a consequence of Theorem 3.1 the determinant and the adjoint of the matrices A(B_{0} + B_{1}), (B_{0} + B_{1})A ∈ M_{n}(R_{0}[d_{1}]) are defined: det(A(B_{0} + B_{1})) is called the right determinant (with respect to the given central ℤ_{2}-grading R = R_{0} ⊕ R_{1}) and (B_{0} + B_{1})adj(A(B_{0} + B_{1})) is called the right adjoint (with respect to the given central ℤ_{2}-grading R = R_{0} ⊕ R_{1}) of the matrix A ∈ M_{n}(R). We use the following notations:

rdet(A) = det(A(B_{0} + B_{1})) and radj(A) = (B_{0} + B_{1})adj(A(B_{0} + B_{1})).

Since A(B_{0} + B_{1})adj(A(B_{0} + B_{1})) =det(A(B_{0} + B_{1}))I in M_{n}(R_{0}[d_{1}]), we immediately obtain (in M_{n}(R)) that:

A radj(A) = rdet(A)I.

3.2. Proposition. (i) If T ∈GL_{n}(R_{0}) is an invertible matrix and A ∈ M_{n}(R), then rdet(T AT^{−1}) =rdet(A) and radj(T AT^{−1}) = T (radj(A))T^{−1}. (ii) If A ∈ M_{n}(R_{0}), then rdet(A) = (det(A))^{n} and radj(A) = (det(A))^{n−1}adj(A).

Proof. (i) In view of T A_{0}T^{−1} ∈ M_{n}(R_{0}) and T A_{1}T^{−1} ∈ M_{n}(R_{1}), the companion matrix of T AT^{−1} = T A_{0}T^{−1} + T A_{1}T^{−1} is T A_{0}T^{−1} + T A_{1}T^{−1}x. Using adj(A_{0} + A_{1}x) = B_{0} + B_{1}x, we obtain that

adj(T A_{0}T^{−1} + T A_{1}T^{−1}x) = adj(T (A_{0} + A_{1}x)T^{−1}) = T (adj(A_{0} + A_{1}x))T^{−1} =
= T (B_{0} + B_{1}x)T^{−1} = (T B_{0}T^{−1}) + (T B_{1}T^{−1})x.

It follows that

rdet(T AT^{−1}) = det(T AT^{−1}(T B_{0}T^{−1} + T B_{1}T^{−1})) =
= det(T A(B_{0} + B_{1})T^{−1}) = det(A(B_{0} + B_{1})) = rdet(A)

and

radj(T AT^{−1}) = (T B_{0}T^{−1} + T B_{1}T^{−1})adj(T AT^{−1}(T B_{0}T^{−1} + T B_{1}T^{−1})) =
= T (B_{0} + B_{1})T^{−1}adj(T A(B_{0} + B_{1})T^{−1}) =
= T (B_{0} + B_{1})T^{−1}T (adj(A(B_{0} + B_{1}))) T^{−1} = T (radj(A))T^{−1}.

(ii) Since A ∈ M_{n}(R_{0}) implies that A_{0} = A and A_{1} = 0, from adj(A_{0} + A_{1}x) = B_{0} + B_{1}x we get that B_{0} =adj(A) and B_{1} = 0. Thus

rdet(A) = det(A(B_{0} + B_{1})) = det(A_{0}B_{0}) = det(det(A)I) = (det(A))^{n}

and

radj(A) = (B_{0} + B_{1})adj(A(B_{0} + B_{1})) = B_{0}adj(A_{0}B_{0}) =
= adj(A)adj(det(A)I) = adj(A)(det(A))^{n−1}I = (det(A))^{n−1}adj(A).  □

If (R_{0}, R_{1}) is a ℤ_{2}-grading of the ring R, then (R_{0}[t], R_{1}[t]) is a natural ℤ_{2}-grading of the polynomial ring R[t] of the commuting indeterminant t. It is straightforward to see that (R[t])[x, σ_{t}]/(x^{2}) ≅ (R[x, σ]/(x^{2}))[t] and \overline{R[t]} = (R_{0}[t]) ⊕ (R_{1}[t])x ≅ (R_{0} ⊕ R_{1}x)[t] = \overline{R}[t] are ring isomorphisms, where σ_{t} : R[t] ⟶ R[t] is the natural extension of σ. For a central ℤ_{2}- grading (R_{0}, R_{1}), the induced ℤ_{2}-grading (R_{0}[t], R_{1}[t]) of R[t] is also central: R_{0}[t] ⊆ Z(R[t]).

We define the right characteristic polynomial (with respect to the given central ℤ_{2}-grading R = R_{0} ⊕ R_{1}) of a matrix A ∈ M_{n}(R) as the right determinant (with respect to the induced central ℤ_{2}-grading R[t] = R_{0}[t] ⊕ R_{1}[t]) of the matrix tI − A ∈ M_{n}(R[t]), where I is the identity matrix in M_{n}(R):

χ_{A}(t) = rdet(tI − A) = λ_{0} + λ_{1}t + ⋯ + λ_{k}t^{k} ∈ R[t], λ_{0}, λ_{1}, …, λ_{k} ∈ R and λ_{k} ≠ 0.

Since GL_{n}(R_{0}) ⊆GL_{n}(R_{0}[t]), an immediate consequence of Proposition 3.2 is that χ_{T AT^{−1}}(t)= χ_{A}(t) for any invertible matrix T ∈GL_{n}(R_{0}).

3.3. Proposition. If χ_{A}(t) = λ_{0} + λ_{1}t + ⋯ + λ_{k}t^{k} is the right characteristic polynomial of the n × n matrix A ∈ M_{n}(R), then k = n^{2} and λ_{n^{2}} = 1, λ_{0} = rdet(−A).

Proof. If A = A_{0} +A_{1} with A_{0} ∈ M_{n}(R_{0}) and A_{1} ∈ M_{n}(R_{1}), then tI −A = (tI −A_{0})+(−A_{1}) with tI − A_{0} ∈ M_{n}(R_{0}[t]) and −A_{1} ∈ M_{n}(R_{1}[t]). The companion matrix of tI − A in M_{n}(\overline{R[t]}) ≅ M_{n}(\overline{R}[t]) is (tI − A_{0}) + (−A_{1})x = tI − (A_{0} + A_{1}x) (here \overline{R[t]} ≅ \overline{R}[t] is a commutative ring). It is well known that each of the elements in the diagonal of adj(tI − (A_{0} + A_{1}x)) is a polynomial in \overline{R}[t] with leading term t^{n−1}. The non-diagonal entries in adj(tI − (A_{0} + A_{1}x)) are polynomials in \overline{R}[t] of degree less than n − 1. In consequence, the matrices B_{0}(t) ∈ M_{n}(R_{0}[t]) and B_{1}(t) ∈ M_{n}(R_{1}[t]) in

adj((tI − A_{0}) + (−A_{1})x) = B_{0}(t) + B_{1}(t)x

have the following properties: each non-diagonal entry of B_{0}(t) and each entry of B_{1}(t) is of degree (in t) less than n − 1, moreover the leading term of each diagonal element in B_{0}(t) is t^{n−1}. Thus each element in the diagonal of the product matrix (tI − A)(B_{0}(t) + B_{1}(t)) is a polynomial with leading term t^{n}. Since the non-diagonal entries in (tI −A)(B_{0}(t)+B_{1}(t)) are of degree less or equal than n − 1, we obtain that the leading term of the right characteristic polynomial det((tI −A)(B_{0}(t)+B_{1}(t))) =rdet(tI −A) = χ_{A}(t) is (t^{n})^{n} = t^{n^{2}}, i.e. that k = n^{2} and λ_{n^{2}} = 1.

To prove λ_{0} =rdet(−A), let adj(−A_{0} − A_{1}x) = C_{0} + C_{1}x with C_{0} ∈ M_{n}(R_{0}) and C_{1} ∈ M_{n}(R_{1}). Now

adj(tI − (A_{0} + A_{1}x)) = (C_{0} + C_{1}x) + C(t)t

for some C(t) ∈ M_{n}(\overline{R}[t]), whence we get that B_{0}(t) + B_{1}(t) = (C_{0} + C_{1}) + H(t)t for some H(t) ∈ M_{n}(R[t]). It follows, that

χ_{A}(t) = rdet(tI − A) = det((tI − A)(B_{0}(t) + B_{1}(t))) =
= det(H(t)t^{2} − AH(t)t + C_{0}t + C_{1}t − A(C_{0} + C_{1})).

Since A(C_{0}+C_{1}) does not contain t, we deduce that the constant term in χ_{A}(t) is rdet(−A) = det(−A(C_{0} + C_{1})). □

3.4. Theorem. If χ_{A}(t) ∈ R[t] is the right characteristic polynomial of an n × n matrix A ∈ M_{n}(R) over a centrally ℤ_{2}-graded ring R = R_{0} ⊕ R_{1} and h(t) ∈ R[t] is arbitrary, then the left substitution of A into the product polynomial χ_{A}(t)h(t) = μ_{0} + μ_{1}t + ⋯ + μ_{m}t^{m} is zero: Iμ_{0} + Aμ_{1} + ⋯ + A^{m}μ_{m} = 0.

Proof. Using

(tI − A)(U_{0} + U_{1}t + ⋯ + U_{m−1}t^{m−1}) = (μ_{0} + μ_{1}t + ⋯ + μ_{m}t^{m})I

in M_{n}(R[t]) ≅ (M_{n}(R))[t] with (radj(tI−A))h(t) = U_{0}+U_{1}t+⋯+U_{m−1}t^{m−1} and U_{i} ∈ M_{n}(R) for the indices 0 ≤ i ≤ m − 1, we can proceed as in the proof of Theorem 4.2 in [4]. □

4. The inverse formula for n × n matrices over the Grassmann algebra

An element g of E = F ⟨v_{1}, v_{2}, …, v_{i}, … | v_{i}v_{j} = −v_{j}v_{i} for all integers 1 ≤ i ≤ j⟩ can be uniquely written in the form

g = c_{g} + ∑_{1≤i_{1}<i_{2}<...<i_{k}}c_{g}(i_{1}, i_{2}, …, i_{k})v_{i_{1}}v_{i_{2}} … v_{i_{k}} ,

where c_{g}, c_{g}(i_{1}, i_{2}, …, i_{k}) ∈ F. Now γ(g) = c_{g} defines an F -algebra homomorphism γ : E → F and γ naturally extends to an F -algebra homomorphism \overline{γ} : M_{n}(E) → M_{n}(F) of the matrix algebras. If N = A − \overline{γ}(A), then it is easy to see that BN is a nilpotent matrix for all B ∈ M_{n}(E). The existence of the inverse matrix (\overline{γ}(A))^{−1} in M_{n}(F) implies the existence of the inverse of A = \overline{γ}(A)(I + (\overline{γ}(A))^{−1}N) in M_{n}(E):

A^{−1} = (I + (−(\overline{γ}(A))^{−1}N) + (−(\overline{γ}(A))^{−1}N)^{2} + ⋯ + (−(\overline{γ}(A))^{−1}N)^{m−1})(\overline{γ}(A))^{−1},

where m is the index of the nilpotence of (\overline{γ}(A))^{−1}N. Thus det(\overline{γ}(A)) ≠ 0 implies the existence of A^{−1} ∈ M_{n}(E). On the other hand, AB = I in M_{n}(E) implies that \overline{γ}(A)\overline{γ}(B) = \overline{γ}(AB) = \overline{γ}(I) = I in M_{n}(F), whence we get that det(\overline{γ}(A)) ≠ 0. In consequence, the existence of A^{−1} in M_{n}(E) is equivalent to det(\overline{γ}(A)) ≠ 0.

4.1. Theorem. For a matrix A ∈ M_{n}(E) we have A = A_{0} + A_{1} for some unique A_{0} ∈ M_{n}(E_{0}) and A_{1} ∈ M_{n}(E_{1}). If A is invertible, then

A^{−1} = (adj(A_{0}) + α_{1}(A))adj (A(adj(A_{0}) + α_{1}(A))) {det (A(adj(A_{0}) + α_{1}(A)))}^{−1},

where adj(A_{0} + A_{1}x) = B_{0} + B_{1}x in M_{n}(\overline{E}) with B_{0} =adj(A_{0}) ∈ M_{n}(E_{0}), B_{1} = α_{1}(A) ∈ M_{n}(E_{1}) and det(A(adj(A_{0}) + α_{1}(A))) is an invertible element of E.

Proof. In view of \overline{γ}(A_{1}) = \overline{γ}(B_{1}) = 0, \overline{γ}(A_{0}) = \overline{γ}(A) and det(\overline{γ}(A)) ≠ 0, we can write that

γ(rdet(A)) = γ(det((A_{0} + A_{1})(B_{0} + B_{1}))) = det(\overline{γ}((A_{0} + A_{1})(B_{0} + B_{1}))) =
= det(\overline{γ}(A_{0} + A_{1})\overline{γ}(B_{0} + B_{1})) = det(\overline{γ}(A_{0})\overline{γ}(B_{0})) = det(\overline{γ}(A_{0}B_{0})) =
= γ(det(A_{0}B_{0})) = γ(det(det(A_{0})I)) = γ((det(A_{0}))^{n}) =
= (γ(det(A_{0})))^{n} = (det(\overline{γ}(A_{0})))^{n} = (det(\overline{γ}(A)))^{n} ≠ 0,

whence we get that rdet(A) is an invertible element of E. From A radj(A) =rdet(A)I, the right multiplication by (rdet(A))^{−1} gives that A^{−1} =radj(A)(rdet(A))^{−1}, where radj(A) = (B_{0} + B_{1})adj(A(B_{0} + B_{1})) and rdet(A) =det(A(B_{0} + B_{1})). □

4.2. Remark. The idea of considering the companion matrix A_{0}+A_{1}x arose in the following way. If A ∈ M_{n}(E) with A = A_{0} +A_{1} and v_{i} is a generator of E not occurring in the elements of A, then A can be completely read off the matrix A_{0} + A_{1}v_{i} and A_{0} + A_{1}v_{i} ∈ M_{n}(E_{0}) lies in a commutative environment. Thus the use of A_{0} + A_{1}v_{i} instead of A is a natural challenge.

References

[1] Domokos, M.: Cayley-Hamilton theorem for 2×2 matrices over the Grassmann algebra. J. Pure Appl. Algebra 133 (1998), 69–81. Zbl 0941.15026

[2] Kemer, A. R.: Varieties of Z_{2}-graded algebras. Math. USSR Izv. 25 (1985), 359–374. Zbl 0586.16010

[3] Kemer, A. R.: Ideals of Identities of Associative Algebras. Translations of Math. Monographs, Vol. 87 (1991), AMS Providence, Rhode Island. Zbl 0732.16001

[4] Szigeti, J.: New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings. Proc. Amer. Math. Soc. 125(8) (1997), 2245–2254. Zbl 0888.16011

Received September 25, 2000