摘要
Suppose F is a field, and n, p are integers with 1 ≤ p 〈 n. Let Mn(F) be the multiplicative semigroup of all n × n matrices over F, and let M^Pn(F) be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of Mn(F) such that F M^Pn(F). A map f : F→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 Mn(F)) is described.
Suppose F is a field, and n, p are integers with 1 ≤ p 〈 n. Let Mn(F) be the multiplicative semigroup of all n × n matrices over F, and let M^Pn(F) be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of Mn(F) such that F M^Pn(F). A map f : F→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 Mn(F)) is described.
基金
the Chinese NSF under Grant No.10271021
the Younth Fund of Heilongjiang Province
the Fund of Heilongjiang Education Committee for Oversea Scholars under Grant No.1054HQ004