# Tagged Questions

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### Short exact sequences and finite injective dimension

Say that $0 \to M \to N \to L \to 0$ is a short exact sequence of modules in a Noetherian local ring and that inj dim$(M)$, inj dim$(N) < \infty$. Does this imply that $L$ also have finite ...
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### Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...
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### Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
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### Hochschild homology with trivial coefficients: how to make $K$ an $M_n(K)$-module

Let $R$ be a ring, $A$ an associative $R$-algebra, and $M$ an $A$-$A$-bimodule. Then the Hochschild homology of $A$ with coefficients in $M$, denoted $HH_\ast(A)$, is the homology of the chain complex ...
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### On maximal submodules of projective modules

I know that any non-zero projective module has a maximal submodule. But is it true that any proper submodule is contained in a maximal submodule !?
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### On a particular $K[x,y]$-module

This is a follow up from HERE. Suppose $K$ is a field and consider $K$ as a $K[x,y]-$module where the scalar product is defined by $f(x,y)\cdot k = f(0,0)\cdot k$. Is $K$ injective or flat as ...
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### Is it true that Tensor product of injective modules is injective?

Is it true that if $M$, $N$ are injective modules over a commutative ring $R$ (with identity) then $M\otimes_R N$ is also injective ?
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### Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
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### When do we have $m\otimes n = 0$ [duplicate]

Let $M$ and $N$ be $R$-modules ($R$ a commutative ring with identity). Let $m \in M$ and $n \in N$. Is there any necessary and sufficient condition to have $m\otimes n = 0$ (as an equation in ...
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### A module with 300 elements

I have got this problem. Let it be $R=M_{2}(Z)$ the ring of square matrices over the integers. I need to find a $R-$module with $300$ elements and one question for this problem, can be there a ...
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### Definition of (co)homology of groups and Lie algebras: actions and augmentations

In the Chevalley-Eilenberg chain complex, what is $ux_i$? What does "trivial $\frak{g}$-module $k$" mean? Below I denote $R=k$ (any commutative unital ring). How is the augmentation (last map in the ...
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### Four term exact sequence for Artin algebras

Suppose $\Lambda$ is an Artin algebra, i.e. an algebra finitely generated as $R$-module for a commutative Artin ring $R$, $A$ is a finitely generated left $\Lambda$-module, and $X$ a finitely ...
Let $R$ be a ring with identity, $I$ an ideal and $M$ a left injective module with $IM= 0$. How can I show that $M$ is an injective $\frac RI$ module?