On the Preservation of Exactness from Module Sequences to Corresponding Sequences of Semimodules of Submodules

Abstract

Let R be a commutative ring with identity, and I(R) denote the semiring of ideals of R. Moreover, let S(M) be the semimodule of submodules of a unitary R-module M. It is shown that S(−), which assigns to each R-module homomorphism f : M → M′, the I(R)-semimodule homomorphism S(f ) : S(M) → S(M′) defined by S(f )(N) = f (N), acts as a covariant functor. In this paper, we examine the preservation of several types of morphisms by S(−). In particular, the preservation of exact module sequences by S(−), and the hom-functors Hom(S(L), S(−) and Hom(S(−), S(L), for any R-module L, are investigated.