Some connections between $R$-modules and $S$-acts via the $k$-realization functor

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, ET.C, Islamic Azad University, Tehran, Iran.

2 Department of Mathematics, SR.C., Islamic Azad University, Tehran, Iran.

3 Department of Mathematics, ET.C., Islamic Azad University, Tehran, Iran.

Abstract

For a commutative pointed monoid $S$ and a commutative unital ring $k$, let $k[S]$ be the commutative ring consisting of finite $k$-linear sums of non-zero elements of $S$. In this paper, we investigate some properties of the $k$-realization functor from the category $S$-{\bf Act}$_{0}$ of all pointed $S$-acts to the category $k[S]$-{\bf Mod} of all $k[S]$-modules, which is a left adjoint to the forgetful functor. Using this adjunction, we show that $S$-{\bf Act}$_{0}$ has enough injective objects. Finally, we prove that the functor $k[-]$ is faithful but not full.

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References

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Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 23 June 2026

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