Derivations of rings of infinite matrices

We describe derivations of several important associative and Lie rings of infinite matrices over general rings of coefficients.


Introduction
Let R be an associative ring with unit 1 R , and let I be an infinite set.Consider the ring M(I, R) of (I × I)-matrices over the ring R having finitely many nonzero entries in each column.The ring M(I, R) is isomorphic to the ring of endomorphisms of a free R op -module of rank card(I).Here, R op is a ring that is anti-isomorphic to R; and card(I) is the cardinality of the set I.
Consider also the subring M ∞ (I, R) < M(I, R) of all (I ×I)-matrices over R having finitely many nonzero entries, and the subring M rcf (I, R) < M(I, R) of all (I × I)-matrices over R having finitely many nonzero entries in each row and in each column.
Recall that an additive mapping d : R → R is called a derivation if The purpose of this paper is to determine derivations of the rings M ∞ (I, R), M rcf (I, R), M(I, R).Recall that all derivations of a ring form a Lie ring; see [6].
Every derivation of the ring R gives rise to a derivation of the ring M(I, R) that leaves M rcf (I, R) and M ∞ (I, R) invariant.Hence, the Lie ring Der(R) lies in each of the Lie rings Der(M(I, R)), Der(M rcf (I, R)), Der(M ∞ (I, R)).
For an element a ∈ M(I, R), let In [4], we proved Theorem 1 in the case when R is a field and the derivation d is linear.W. Ho lubowski and S. Żurek proved Theorem 1 under the assumptions that (i) the set I is countable, (ii) the ring R is commutative, and (iii) the derivation d is R-linear; see [5].
An arbitrary associative ring R gives rise to the Lie ring Using the proof of Herstein's Conjectures by K.I.Beidar, M. Brešar, M.A. Chebotar and W.S. Martindale (see [1][2][3]) and Theorem 1, we obtain descriptions of derivations of Lie rings under the assumption that For an element a ∈ R and indices i, j ∈ I let e ij (a) denote the (I × I)-matrices having the element a at the intersection of the i-th row and the j-th column, all other entries are equal to zero.
The following lemma is straightforward.
Let Z be the ring of integers.Then Z • 1 R is a subring of the ring R. The ring M(I, R) is a bimodule over the ring M ∞ (I, Z • 1 R ).

Lemma 2. For an arbitrary derivation
for an arbitrary index k ∈ I.
By Lemma 1, a kl ) k,l∈I = 0.The (p, q)-entry of the matrix on the left hand side is The diagonal entries of the matrix [X, e kk (1 R )] are equal to zero. Define ij for i = j; a ii = 0; and v = (a ij ) i,j∈I .By the above, The j-th column of the matrix v is equal to the j-th column of the matrix d e kk (1 R ) ∈ M(I, R).Hence, only finitely many entries in the j-th column are different from zero, hence v ∈ M(I, R).This completes the proof of the lemma.

d
(ab) = d(a)b + ad(b) for arbitrary elements a, b ∈ R. Let V be a bi-module over a ring R.An additive mapping d : R → V is called a derivation or a 1-cocycle if d(ab) = d(a)b + ad(b) for arbitrary elements a, b ∈ R. For an element v ∈ V the mapping d v : R → V, d v (a) = av − va is a derivation.
ad(a) : x → [a, x] = ax − xa denote the inner derivation.Since M ∞ (I, R) is a two-sided ideal in M rcf (I, R), it follows that M ∞ (I, R) is invariant under any inner derivation ad(a), a ∈ M rcf (I, R).Theorem 1.(a) An arbitrary derivation d of the ring M ∞ (I, R) is of the type d = ad(a) + u, where a ∈ M rcf (I, R), u ∈ Der(R); (b) an arbitrary derivation d of the ring M(I, R) (resp.M rcf (I, R)) is of the type d = ad(a) + u, where a ∈ M(I, R) (resp.a ∈ M rcf (I, R)), u ∈ Der(R).
1 2 ∈ R. Theorem 2. (a) An arbitrary derivation d of the Lie ring sl ∞ (I, R) is of the type d = ad(a) + u, where a ∈ gl rcf (I, R), u ∈ Der(R); (b) an arbitrary derivation d of the Lie ring gl(I, R) (resp.gl rcf (I, R)) is of the type d = ad(a) + u, where a ∈ gl(I, R) (resp.a ∈ gl rcf (I, R)), u ∈ Der(R).

Proof.
The assertion (b) immediately follows from the fact that M ∞ (I, R) is a two-sided ideal in the ring M rcf (I, R).Let v = (a ij ) i,j∈I ∈ M(I, R), v, e kk (1 R ) ∈ M rcf (I, R) for any k ∈ I.All entries in the k-th row of the matrix v, except for the diagonal one, are negatives of the corresponding entries in the k-th row of the matrix [v, e kk (1 R )].Since v, e kk (1 R ) ∈ M rcf (I, R), it follows that only finitely many entries of the k-th row of the matrix v are different from zero.Hence, v ∈ M rcf (I, R).This completes the proof of the lemma.Now, we are ready to prove Theorem 1. Proof.Let d be a derivation of the ring M ∞ (I, R).By Lemma 2, there exists a matrix v ∈ M(I, R) such that d ′ = d − d v maps all matrix units e kk (1 R ) to zero.This implies that d ′ e ij (R) ⊆ e ij (R) for all indices i, j ∈ I. Define the additive mapping d ij : R → R via d ′ e ij (a) = e ij d ij (a) .Clearly, d ij (1 R ) = 0. Let a, b ∈ R; i, j, k ∈ I.We have e ij (ab) = e ik (a)e kj (b).Hence, d ij (ab) = d ik (a) • b + a • d kj (b).Let b = 1 R .Then d ij (a) = d ik (a).This implies that d ij does not depend on i, j.Define d = d ij ; i, j ∈ I.The mapping d is a derivation of the ring R. Hence, d ′ ∈ Der(R).All derivations from Der(R) map M ∞ (I, R) into itself.Therefore, d v M ∞ (I, R) ⊆ M ∞ (I, R).By Lemma 3(a), it implies that v ∈ M rcf (I, R) and completes the proof of the first part of the theorem.Let d be a derivation of the ring M(I, R).Arguing as above, we find a matrix v ∈ M(I, R) such that u = d − d v ∈ Der(R).If d is a derivation of the ring M rcf (I, R), then the mapping d v maps M ∞ (I, R) to M rcf (I, R).By Lemma 3(a), v ∈ M rcf (I, R).This completes the proof of the theorem.