Derivations and Automorphisms of Certain Subrings of Matrix Rings

Abstract

Let K be an arbitrary associative ring with identity. We denote by Mn(K) the ring of all nxn matrices over K. Say K=F for some field F. Then it is a consequence of Skolem-Noether theorem that every automorphism of Mn(F) is inner.

Recall that a derivation of a ring K is an additive map D:KK satisfying D(ab)=D(a)b+aD(b) for all a,b in K.

The problem of describing all derivations of a ring is an interesting topic for many researchers. Many papers are concerned with the study of derivations of matrix rings and their subrings.

As a result of Skolem-Noether theorem, every derivation of Mn(F) is inner.

In 1982, S.A. Amitsur proved that any derivation of Mn(K) is the sum of an inner derivation and a derivation arising from a derivation on K where K is an arbitrary ring.

Let NTn(K) be the ring of all niltriangular nxn matrices over K whose entries are all zeros on and above the main diagonal. V.M. Levchuk characterized the automorphisms of NTn(K) in 1983.

In 2006, J.H. Chun and J.W. Park proved that every derivation of NTn(K) is a sum of a certain diagonal, trivial extension and a strongly nilpotent derivation.

The set defined by Rn(K,J)=NTn(K)+Mn(J) is a ring with usual matrix addition and multiplication where K is a unital ring and J is an ideal of K. The automorphisms of the ring Rn(K,J) were described by F. Kuzucuoğlu and V.M. Levchuk under certain spesific properties.

In the first section of this thesis, we give the historical background of derivations and automorphisms of some certain matrix rings and algebras.

In the second section, we characterize all derivations of Rn(K,J).

Recall that the Jordan multiplication on a ring K is given with a.b=ab+ba for any a,b in K. An additive map \Omega of K satisfying \Omega(a.b)=\Omega(a).b+a.\Omega(b) is called a Jordan derivation of K.

Every derivation is a Jordan derivation but there are Jordan derivations which are not derivations.

All Jordan derivations of the ring NTn(K) are described by F. Kuzucuoğlu in 2011.

For an arbitrary associative and 2-torsion free ring K with identity and an ideal J of K, we describe all Jordan derivations of Rn(K,J) in the third section.