# Tagged Questions

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

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### Modules over algebras vs Modules over Rings?

Since I've started module theory I was confused with a point. What is more general, the theory of modules over algebras or over arbitrary rings? I hope this is not a pointless question so let me try ...
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### On selfinjectivity of Hopf algebras

Any group algebra $kG$ of a finite group is selfinjective. More generally Gentile proves that for a group ring $RG$ with $R$ commutative and torsion free as a $\Bbb Z$-module, $RG$ is selfinjective ...
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### Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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### Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
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### Let $M$ be a finitely generated module over a P.I.D.. If $M=A\oplus B$, $A$ is a torsion submodule and $B$ is free, then $A=M_{\text{tor}}$.

I am reading the Goodman's Algebra. There is a lemma. Let $M$ be a finitely generated module over a principal ideal domain $R$. (a) If $M=A\oplus B$, $A$ is a torsion submodule, and $B$ is free, ...
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### $\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N)$

Let $(M_i)_{i\in I}$ be a collection of $R$-moduls. Show that for all $N\in \text{Ob}(_R\text{Mod})$ is $$\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N).$$ My ...
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### Expressing an $R$-module as a direct sum

Let $R$ be a ring with $1$, $M$ a right $R$-module, and $f:M \to R$ a homomorphism of $R$-modules with $f(M)=R$. Then there is a decomposition $M=K \oplus L$ such that $f \vert_L:L\to R$ is an ...
### Finding the structure of an $F_p[X]$module
$p$ is a prime and $M$ is an $F_p[X]-$ module. Given $(X-1)^3M=0, \vert (X-1)^2M\vert=p, \vert (X-1)M\vert=p^3$ and $\vert M\vert =p^7$, determine $M$ as an $F_p[X]-$module, up to isomorphism. Using ...