# Tagged Questions

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

38 views

### Expressing an $R$-module as a direct sum

Let $R$ be a ring with $1$, $M$ a right $R$-module, and $f:M \to R$ a homomorphism of $R$-modules with $f(M)=R$. Then there is a decomposition $M=K \oplus L$ such that $f \vert_L:L\to R$ is an ...
53 views

### Finding the structure of an $F_p[X]$module

$p$ is a prime and $M$ is an $F_p[X]-$ module. Given $(X-1)^3M=0, \vert (X-1)^2M\vert=p, \vert (X-1)M\vert=p^3$ and $\vert M\vert =p^7$, determine $M$ as an $F_p[X]-$module, up to isomorphism. Using ...
47 views

36 views

### -3 is quadratic residue if and only if $p \equiv 1,7 \pmod {12}$ [closed]

I have to prove that -3 is quadratic residue if and only if $$p \equiv 1,7 \pmod {12}$$ I know one method (with symbol Legendre'a) but I don't get. If someone can explain me I will be happy or give ...
34 views

### Prove that $3^{(q-1)/2} \equiv -1 \pmod q$ then q is prime number.

$q=2^m+1, m\ge 2$. Prove that if $$3^{(q-1)/2} \equiv -1 \pmod q$$ then q is prime number. I want to use if $q-1 | \phi(q)$, then q is prime number. But I don't know how to transform above equation. ...
32 views

30 views

### Induced Representations from Tensor Product of Modules

In Chapter 7 of J P Serre's book on Representation Theory, he defines a $\mathbb{C}[G]$-module as $W'= \mathbb{C}[G]\otimes_{\mathbb{C}[H]}W$ and claims it to be the $\mathbb{C}[G]$-module obtained ...
12 views

### the socle is semisimple

The socle of a module $M$ is semisimple. I know that the socle is $soc(M)=\sum\{N\leq M| \text{$N$is a simple submodule of$M$}\}$. So $soc(M)=N_1+\cdots+N_k$, where $N_j$ are simple submodules, for ...
15 views

### for a projective module $P$ we have $J(R)P$ is superfluous in $P$ if and only if $\frac{P}{J(R)P}$ has a projective cover.

Let $P$ be a projective module, then $J(R)P$ is superfluous in $P$ if and only if $\frac{P}{J(R)P}$ has a projective cover. Here $J(R)$ is the jacobson radical of $R$ (the underlying ring).
14 views

### Proving $R$ is semisimple if and only if $\frac{R}{J(R)}$ is semiperfect and $R(J)$ and every quotient of $R$ has non-trivial right Socle.

I must prove that a ring $R$ is semiperfect if and only if the following two conditions hold: $\frac{R}{J}$ is semisimple. Every quotient of $R$ has non-trivial Socle. I know that $R$ is ...
15 views

### Any artinian ring is coherent and perfect.

Let $R$ be an artinian ring. We must prove that $R$ is also a coherent $R$ module. In other words that every finitely generated ideal is finitely presented. To show that $R$ is perfect I think we ...
59 views

### Can you prove that modules over a field are vector spaces in a categorical way?

Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category (i.e....
25 views

102 views

### Question concerning modules over a Clifford algebra

Let $R$be a commutative ring with unit element, and $M$ be an $R-$module. Let $f:M \times M \to R$ be a nondegenerate symmetric bilinear quadratic form, and $C(f)$ be the corresponding Clifford ...
I read the book "K-theory for operator algebras" and I have a question about the definition of Kasparov modules for graded $C^*$-algebras $A$ and $B$. Definition: Let $A$ and $B$ graded $C^*$-...
We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ($a\in R,x\in M$) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if \$a=t, ...