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

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Birjand, Birjand, Iran

2 Department of Basic Sciences, Technical and Vocational University (TVU), Tehran, Iran

Abstract

‎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 $0\rightarrow M' \xrightarrow{f} M\xrightarrow{g} M'' \rightarrow 0$ with $M''$ a radical module. In particular, we provide conditions under which the hom-functors preserve short exact sequences of modules. 

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References

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History

  • Receive Date: 02 October 2025
  • Revise Date: 24 February 2026
  • Accept Date: 24 April 2026

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