# Questions tagged [module-isomorphism]

The tag has no usage guidance.

93 questions
Filter by
Sorted by
Tagged with
22 views

• 1,623
12 views

### 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 ...
13 views

### 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 $M$ ...
1 vote
33 views

• 113
85 views

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

93 views

• 265
132 views

1 vote
96 views

### 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 ...
• 1,074
1 vote
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\...