I have a commutative algebra class and I heard the theorem from the professor: Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} E_i$ is also injective. My questions are: How to prove that? My professor does not talk about the proof of it […]

In order to test injectivity of a module $M$ it suffices to check if every linear map from an arbitrary ideal extends to the ring or not. Similarly in order to check the flatness of a module $M$ it suffices to check whether tensoring with it preserves injectivity of $0 \to I \to R$. Is […]

Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How can I show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$, then $M$ is injective?

Let $I$ be uncountable set. How to prove that $\left(\prod\limits_{i\in I}\mathbb{Z}x_i\right)/\sum\limits_{i\in I }\mathbb{Z}x_i$ always contain uncountable injective module ? Can we construct it explicitly ?

$M$ is a flat $R$-module if and only if $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$ is injective. One direction is easy. Suppose $M$ is flat. We know that $$ \hom_\mathbb{Z}(-\otimes M, \mathbb{Q/Z}) \cong_{\mathbb{Z}} \hom_{\mathbb{Z}}(-,\hom_{\mathbb{Z}}(M,\mathbb{Q/Z}))$$ Since $- \otimes M$ is exact and $\mathbb{Q/Z}$ is injective, the left functor is exact which shows that the right functor is exact, i.e. $\hom_{\mathbb{Z}}(M,\mathbb{Q/Z})$ is […]

Suppose that $R$ is a local Noetherian ring. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is Artinian? It is easy if $R$ is Cohen-Macaulay, because we know that if there exists a nonzero finitely generated module $M$ of finite injective dimension then $\mathrm{id}\;M=\mathrm{depth}\;R$. So in […]

Let $R$ be an artinian chain (uniserial) ring. I want to prove that $R$ is self-injective, that is, any $\alpha:I\rightarrow R$ right $R$-module homomorphism can be extended to a homomorphism $\tilde{\alpha}:R\rightarrow R$ where $I$ is a right ideal of $R$. Let $J$ be the jacobson radical of $R$. We knov that all right ideals of […]

It is well-know that flat base change in general does not preserve injective modules, i.e. let $M$ be an injective $R$-module and $R\to S$ be a flat morphism, then it is not necessary that $M\otimes_R S$ is an injective $S$-module. Now let’s consider a more special case: Let $R$ be a commutative noetherian algebra over […]

In the category of $\mathbb{Z}$-modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$? I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is also an essential extension. We already know that $\mathbb{Z}/n\mathbb{Z} \rightarrowtail \prod_{i \in \lbrace 1,\ldots,n \rbrace}(\mathbb{Q}/\mathbb{Z})_i $. But is this an essential extension? I can’t prove or disprove it. Guessing […]

Suppose that $R$ is a integral domain. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is field?

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