Derivations and Automorphisms of Some Infinite Matrix Algebras

Abstract

Let R be a commutative ring with identity and M_n(R) be the algebra (ring) of all n x n matrices over R. Note that an additive D of a ring R into itself is said to be a derivation of R if D(xy)=D(x)y+xD(y) for all x, y ∈ R. Studies on automorphisms and derivations of matrix algebras and matrix rings have been actively continuing since the 1950s. In the first study on this subject, in the case of R being a field, Skolem-Noether showed that each automorphism of the matrix algebra M_n(R) is an inner automorphism. It has also been shown that every derivation of M_n(R) is inner in the case when R is a field. Later on, these studies were extended to the subalgebras (subrings) of the matrix algebra (ring) M_n(R).

Since the 2000s, studies on Lie and Jordan automorphisms and Lie and Jordan derivations of M_n(R) matrix algebras (rings) and subalgebras (subrings) have been started to appear in the literature. This thesis aims to bring automorphism and derivation problems to infinite matrix algebras and rings. The first chapter of this thesis, which consists of five chapters, contains the historical development of the subject of this thesis and relevant information. Second chapter covers some basic definitions and theorems which will help us better understand the work to be done in the following chapters. In the third chapter, infinite matrix algebras and rings are introduced, and some of their basic properties are observed. In the fourth chapter, derivations of column finite matrix rings are discussed. In the last chapter, all Lie derivations of (upper) niltriangular infinite matrix algebras are described.