# Questions tagged [module-isomorphism]

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### Proof for $P(R) \leq I-rad(R_R)$

The (right) $\textbf{isoradical}$ $I-rad(R_R)$ of a ring $R$ is the intersection of the annihilators of all isosimple right $R-$modules where an $\textbf{isosimple}$ module is defined as a non-zero ...
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### Completely virtually semisimple modules are direct sums of isosimple modules.

An $\textbf{isosimple}$ module is defined as a non-zero module whose all non­zero submodules are isomorphic to it. An $R$-module $M$ is called $\textbf{virtually semisimple}$ if every submodule of $M$ ...
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I am preparing myself for an upcoming exam, and I've found the following problem Let $M$ be a $R$-module and $N_1 \subset N_2 \subset M$ be $A$-submodules. Use the Snake Lemma to show that $$\frac{M/... 0 votes 0 answers 26 views ### Submodule of a noetherian module is noetherian [duplicate] Assume R is a ring, \psi:M\rightarrow N a surjective R-modules homomorphism. Then, M is noetherian \iff N and K=\operatorname{Ker}(\psi) are noetherian. So I proved \leftarrow. Now I ... 0 votes 1 answer 36 views ### Does taking covariant Hom commute with taking (co-)kernel? Let A be a Commutative ring and R be a commutative A-algebra. So every R-module has a natural A-module structure, and for every A-module W, and R-module M, the A-module \text{Hom}... 1 vote 0 answers 59 views ### A co-isosimple module is co-cyclic. I have tried to prove the statement that if an R-module is co-isosimple, then it is co-cyclic. In the paper that this stated, this is mentioned as since a co-isosimple module is artinian and uniserial,... 0 votes 2 answers 60 views ### linearly isomorphic submodule I'm currently learning about modules and am pretty new to the topic. I recently found this question: Suppose that R is an integral domain and not a field. Give, with proof, an eample of a proper ... 0 votes 1 answer 99 views ### Understanding example 3 on pg. 369 in Dummit & Foote (3rd edition) Here is the example: (3) In general,$$ \mathbb Z_m \otimes_Z \mathbb Z_n \cong \mathbb Z/d\mathbb Z,$$where d is the g.c.d of the integers m and n. To see this, observe first that$$a \... 86 views

### Proving $r(m + n) = rm + rn$ for a type of scalar multiplication.

Here is the question I want to answer: Let $R \subset S$ be commutative rings and let $M$ be an $R$-module. Then $S \otimes_R M$ is an $R$-module generated by $\{s \otimes m \mid s \in S, m \in M\}.$ ... 50 views

### Understanding how the hint will prove injectivity.

Here is the question I want to solve: Let $M$ be a finitely generated $R$-module. Show that if $f \in \mathrm{End}_R(M)$ is surjective then it is also injective. And here is the hint I got for the ... 156 views

### Proving that $M$ is a simple $R-$module.

Here is the question I want to answer: A module is simple if it is not the zero module and it has no proper nonzero submodule. $(a)$ Let $M$ be an $R-$module. Show that the following conditions are ... 53 views

### Do I have to show that elements of $L$ commutes with elements of $N$ (like in case of direct product) and if so, why?

I want to prove the following $a \Longleftrightarrow d$ in the following questoin: Let $R$ be a commutative ring. For $R-$modules $L,M,N$ show that the following conditions are equivalent.(all ... 81 views

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### Why $M/\mathfrak{a}M \oplus M/\mathfrak{b}M \simeq M/(\mathfrak{a \cap b})M$?
Let $M$ be an $A$-module and let $\mathfrak{a}$ and $\mathfrak{b}$ be coprime ideals of A. I must show that $M/ \mathfrak{a}M \oplus M/ \mathfrak{b}M \simeq M/ (\mathfrak{a \cap b})M$. My attempt is ...
Consider a $R$-module homomorphism $\varphi\colon M\to N$. It is well-known that $\ker\varphi=\{0\}$ iff $\varphi$ is injective (equivalently is monic). However, suppose we only have \$\ker\varphi\...