# Tagged Questions

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

14 views

40 views

### Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
26 views

### Help with proving a fact about injective modules

Let $R$ be a ring with identity. I am trying to show that an $R$-module $A$ is injective if and and only if for every left ideal $L$ of $R$ and every $R$-module homomorphism $g: L \rightarrow A$, ...
27 views

### A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
33 views

41 views

20 views

### What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...
14 views

### Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
34 views

37 views

### Let $M$ be any $A$-module. Is it true that $M\oplus A\cong A\oplus A$ implies that $M\cong A$? [duplicate]

Let $M$ be any $A$-module. Is it true that $M\oplus A\cong A\oplus A$ implies that $M\cong A$? I think it would be true and easy to prove with $A$ a PID, but I don't know in the general case. I ...
45 views

### Why is $0_R0_M$ = $0_M$ in a left $R$-module $M$?
Given a left module $M$ over a ring $R$ where $0_M$ is the additive identity in $M$ and $0_R$ is the additive identity in $R$, I am trying to work out why $r0_M= 0_M$ for all $r \in R$. So far I have ...
Let $M, P$ be two finitely generated $A$-modules ($A$ is a commutative ring with $1$). Let also $P$ be projective. I have to prove that Hom$_A(P,M)$ is finitely generated as $A$-module. Before to ...