Exactness preservation by functors mapping modules to semimodules of varieties of submodules

‎Let R be a commutative ring with identity, and M be an R-module. We denote by \zeta(M) the semimodule consisting of all varieties of submodules of M over the semiring \zeta(R) of varieties of ideals of R. In this article, we introduce two functors from the category of R-modules to the category of \zeta(R)-semimodules and investigate conditions under which these functors preserve the short exact sequences of modules. They are the hom-functors \operatorname{Hom}_{\zeta(R)}(\zeta(P),\zeta(-)) and \operatorname{Hom}_{\zeta(R)}(\zeta(-),\zeta(E)) associated with R-modules P and E, respectively. It is shown that \zeta is exact and both \operatorname{Hom}_{\zeta(R)}(\zeta(P),\zeta(-)) and \operatorname{Hom}_{\zeta(R)}(\zeta(-),\zeta(E)) are left exact on any short exact sequence ۰\rightarrow M' \xrightarrow{f} M\xrightarrow{g} M'' \rightarrow ۰ with M'' a radical module. In particular, we provide conditions under which the hom-functors preserve short exact sequences of modules.

کلیدواژه ها:

Variety of submodule‎ ، ‎Semimodule‎ ، ‎Semimodule homomorphism‎ ، ‎Exact sequence‎ ، ‎Radical submodule