For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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Testing if a submodule is free

This is hopefully a very simple question. In "Groebner Bases in commutative algebra" by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, ...
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A better way of showing the set of all $m\times n$ matrix is an $R$ module.

Let $R$ be a ring and $m$ and $n$ be any positive integers. For any $a\in R$ and $A=(a_{ij})\in M_{m\times n}(R)$ define $aA=(aa_{ij})$. Then prove that $M_{m\times n}(R)$ is an $R$-module. Edited: ...
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Subring of a commutative Noetherian ring

We know that it's possible subring of a commutative Noetherian ring is not Noetherian (for example: Subring of a finitely generated Noetherian ring need not be Noetherian?). But if $S$ be a ...