Homomorphisms between semimodules

Abstract

Let R be a semiring and let M and M’ be left R-semimodules. A function α: M → M’, written as acting on the right, is an R-homomorphism if and only if (m + m’)a = ma + m’α and (rm)α = r(ma) for all r ∈ R and all m, m’ G M: We denote the set of all R-homomorphisms from M to M’ by Hom _R (M,M’). This set is always nonempty since it contains the 0-homomorphism Σ0 : m ↦ 0_m’, If α G Hom _R (M, M’), then the image im(α) = {mα | m ∈ M} of a is a subsemimodule of M’ and the kernel ker(α) = {m G M | ma = 0m’} of α is a subsemimodule of M. Note that the kernel of α may equal {0 _m } even though α is far from being monic. Similarly we define the notion of an R-homomorphism of right i?-semimodules. Such functions are written as acting on the left. An R-homomorphism which is monic and epic is an R-isomorphism. If m ∈ M is idempotent, and if α ∈ Hom _R (M,M’), then ma is an idempotent element of M’;. If N is a subsemimodule of a left R-semimodule M, and if a ∈ Hom _R (M,M’), then the restriction of a to N is clearly an iž-homomorphism from N to M’.