Tensor product of semimodules of radical submodules and exact sequences

Abstract

Let R(R) denote the commutative semiring of radical ideals of a commutative ring with identity, and R(M) denote the R(R)-semimodule consisting of all radical submodules of an R-module M. Moreover, R(−) will be the covariant functor from the category of R-modules R−Mod to the category of R(R) semimodules R(R)−Semod mapping any R-module M to the R(R)-semimodule R(M) and any R-module homomorphism f : M → M totheR(R)-semimodulehomomorphismR(f) : R(M) → R(M)definedby R(f)(N) = rad(f(N)). In this article, we investigate the conditions under which the natural tensor functor R(−)⊗R(R) R(T) (for an R-module T) preserves module exact sequences, by considering a tensor product for semimodules over commutative semirings and an exactness for semimodule sequences similar to those of modules over commutative rings. Among others, it is proved that for any ideal I of an absolutely flat ring R, R(−) ⊗R(R) R(R/I) preserves any short exact sequence of finitely generated faithful multiplication R-modules. Also, it is shown that for any F-vector space W, R(−) ⊗R(F) R(W) preserves any short exact sequence of vector spaces