Some Studies About Automorphisms Of Matrix Algebras Over Semirings

The automorphisms of matrix algebras and their subalgebras are one of important subject in matrix theory. In 1927,Skolem obtained the famous Skolem-Noether theory:the automorphisms of the n × n matrix algebra over a field are inner.Since then,lots of researches in this area have been done.Among these researches,the objectes concerned are matrix algebras over fields or rings.In this paper,we mainly study the automorphisms of matrix algebras over semirings.The thesis is composed of four chapters. In chapter 1,we primarily give some preliminary conceptions and lemmas. In chapter 2,we will discuss the automorphisms of the matrix algebras M n (R) over a commutative semiring R .By using properties of scalar matrices over semirings,we generalize algebraic properties for the automorphisms of matrix algebras over commutative rings and obtain some algebraic properties for the automorphisms of matrix algebras over commutative semirings.By the means of permanents,we show that the nth power of any automorphism of the n× n matrix algebra over a nonnegative semiring is inner. In chapter 3,we will consider the automorphisms of triangular matrix algebras over commutative semirings.By the means of some properties of matrix,we prove that algebra automorphisms of the triangular matrix algebra Tn (R) over a commutative semiring R are inner. In chapter 4,we will characterize multiplicative semigroup automorphisms of the C -algebras of triangular matrices over semirings.We prove that if n ≥2 and R is an effective semiring or a semiring in which all idempotents are central elements,then the mapping Φ of the triangular matrix C -algebra Tn (R) over the semiring R is a multiplicative semigroup automorphism if and only if there exist an invertible matrix G ∈ Tn(R) and a semiring automorphism τ of R such that Φ ( A) =G?1τ(A)G for all A = ( aij)n×n in Tn (R),where τ ( A) = (τ(aij))n×n. The main results obtained in the paper generalize the previous results by Issacs,Kezlan and Cao and Zhang.