Tagged Questions

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

40 views

48 views

On boolean algebras as rings, modules and/or R-algebras

After trying to make sense of first order logic from an algebraic point of view I started to read about boolean algebras (similar to the explanations given here: wikipedia on boolean algebras. I also ...
22 views

Element separated from a submodule by a homomorphism

Let $M$ be a module over a commutative ring $A$, $M'$ a submodule, and $y\in M\setminus M'$. Then $y$ can be separated from $M'$ by homomorphism. What I wish to prove is that there exists an $A$-...
38 views

Socles and factors

Let $A$ be a finite dimensional algebra over a field $K$ and let $M$, $M'$ and $N$ be $A$-modules. Suppose that $M'\subseteq M$ and assume that $N$ has simple socle. Let $f: M \longrightarrow N$ be ...
28 views

About rank of free module compared to spanning and LI set

Let M be a free R-module where R is commutative ring with unity. Q1: If M is FG by n vectors. Will rankM $\leq$ n? Q2: If M has a LI set $S$. Will #$S \leq$ rankM? What if R is PID? Are above ...
27 views

Writing a homomorphism in multiplicative form

Consider the additive group $\mathbb {Z_2}=\{0,1\}$ and consider the group $\mathbb Z_2^n$. Let $u_1,\ldots,u_d$, $(d>n)$ be not necessarily distinct , non-zero elements of $\mathbb Z_2^n$. Define ...
18 views

Can we know what are the form of submodules of Z\oplus Z. [duplicate]

I know for instance Z(2,3), Z(2n,0) or Z(n,n) are submodules of the Z-module Z+Z. But can we know what are the submodules of Z\oplus Z exactly?
15 views

Relation between the dual module and the dual of a vector space.

I need to know if there is a relation between the dual module of a subspace U of a finite dimensional vector space V, looked as a G-module, and the dual vector space of U. In this case, G is a finite ...
54 views

Why is a bilinear map $M \times N \to B$ the same as a homomorphism $M \to Hom_R(N,P)$?

Eisenbud's "Commutative Algebra" states that: It is elementary that a bilinear map $M \times N \to P$ is the same as a homomorphism $M \to Hom_R(N,P)$ While this makes some intuitive sense I keep ...
27 views

A embedding of tensor product over semisimple algebras

Let $R$ be a semisimple Artinian algebra over the complex number field $\mathbb{C}$, that is, $R$ is isomorphic a finite direct product of matrix rings over $\mathbb{C}$. Let $S$ be a ideal of $R$, ...
57 views

Compute the projective dimension of the given $R$-module

Let $$R=\frac{K[[x,y,z]]}{\left<xz,yz\right>}\text{ and } M=\frac{R}{\left<z+\left<xz,yz\right>\right>}.$$ Compute the projective dimension of $M$ as an $R$-module. My attempt ...
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 ...
55 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 ...
52 views

37 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. ...
33 views

33 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

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 ...
16 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 ...
60 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....
Let $\Bbb K$ be a field and $R = \Bbb K[X]$ be the ring of polynomials with coefficients in $\Bbb K$. Let $\cal L_{\Bbb K}$$(V)$ denote, for a finite dimensional $\Bbb K$-vector space $V$, the set of ...
In Atiyah Macdonald, "Introduction to commutative Algebra" it says: Proposition 2.9:i) Let $M' \xrightarrow[]{u}M \xrightarrow[]{v} M'' \rightarrow 0$ be a sequence of A-modules and homomorphisms. ...