Abstract:

Suppose F is a field, and n,p are integers with 1≤ pn( F) be the multiplicative semigroup of all n× n matrices over F, and let M_n^p( F) be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of M_n( F)
such that F\supseteq M_n^p( F). A map f:F \rightarrow R is called a homomorphism if f(AB)=f(A)f(B) for any A,B∈ F. In particular, f is called an endomorphism if F= R. The structure of all homomorphisms from F to R (respectively, all endomorphisms of M_n( F)) is described.

Key words: Homomorphism, endomorphism, multiplicative semigroup of matrices