# Tagged Questions

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

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### Let a,b have the same divisor (content) in an integral domain A. When can I deduce $a/b\in A^\times$?

Given a Noetherian integral domain A and a finitely generated torsion A-module M, we can define the divisor, or content, of M to be $div(M)= \sum_{P, ht(P)=1} \ell(M_P) [P]$, where the sum ranges ...
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### Module is isomorphic to a direct sum of modules, then the length(M)=sum length(M_i) [closed]

Is there any proof (or even a counterexample) for: If $M\tilde{=} M_1\oplus ... \oplus M_n$, then it follows for the finite length $l(M) = l(M_1)+...+l(M_n)$. (Modules of a commutative ring with ...
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### If every cyclic $R$-module is projective, then $R$ is semisimple.

If every cyclic $R$-module is projective, then $R$ is semisimple. I know how to prove this if the definition of cyclic module is $M=Rm$ where $m$ is an element of $M$. The problem is, we defined in ...
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### What exactly is the $O_X$-module and the corresponding sheaf of modules?
I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...