Let \(\mathcal {R}(R)\) denote the commutative semiring of radical ideals of a commutative ring with identity, and \(\mathcal {R}(M)\) denote the \(\mathcal {R}(R)\) -semimodule consisting of all radical submodules of an R-module M. Moreover, \(\mathcal {R}(-)\) will be the covariant functor from the category of R-modules \({{R}{-}Mod}\) to the category of \(\mathcal {R}(R)\) -semimodules \({\mathcal {R}(R){-}Semod}\) mapping any R-module M to the \(\mathcal {R}(R)\) -semimodule \(\mathcal {R}(M)\) and any R-module homomorphism \( f:M\rightarrow M'\) to the \(\mathcal {R}(R)\) -semimodule homomorphism \(\mathcal {R}(f): \mathcal {R}(M)\rightarrow \mathcal {R}(M')\) defined by \(\mathcal {R}(f)(N)=\operatorname {rad}(f(N))\) . In this article, we investigate the conditions under which the natural tensor functor \(\mathcal {R}(-)\otimes _{\mathcal {R}(R)} \mathcal {R}(T)\) (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, \(\mathcal {R}(-)\otimes _{\mathcal {R}(R)} \mathcal {R}(R/I)\) preserves any short exact sequence of finitely generated faithful multiplication R-modules. Also, it is shown that for any F-vector space W, \(\mathcal {R}(-)\otimes _{\mathcal {R}(F)} \mathcal {R}(W)\) preserves any short exact sequence of vector spaces.
