Tagged Questions

2
votes
0answers
38 views

Structure Theorem for finitely generated modules over PIDs [duplicate]

Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either $$\begin{pmatrix} 0&-1&0&0\\ ...
3
votes
1answer
43 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
1
vote
1answer
42 views

Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
2
votes
0answers
64 views

Help understanding a comment about isomorphisms of two direct sums.

I've recently noticed a few different posts asking about this problem Isomorphism of two direct sums, all using the CRT. I think I understand it this way. What caught my eye is in the above link a ...
2
votes
1answer
80 views

Modules over Principal Ideal Domains

Let $a$ and $b$ be two nonzero elements of a PID $R$. Prove that direct sum $R/(a)\oplus R/(b)$ is isomorphic to the direct sum $R/(u)\oplus R/(v)$, where $u=\gcd(a,b)$ and ...
0
votes
0answers
44 views

Why Syz algorithm for module have dimension problem, how to divide vector and multiply vector at final step to get 1*6 Vector

refer to page 162 in an introduction to grobner basis, william W. Adams https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!338 https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!339 when ...
0
votes
0answers
45 views

How to calculate Grobner basis for Syz(i,f1,f2,g1,g2)

refer to page 174 in an introduction to grobner bases, william A. Adams no matter choose first row or second row of i, f1, f2, g1, g2, the maple code not the same as in the book ...
1
vote
0answers
71 views

Injective hull commutes with Hom

Notation: $E_R(M)$ is the injective hull of $M$. Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ an $R$-module. Then $$\mathrm{Hom}_R(R/I, E_R(M)) \cong E_{R/I}(\mathrm{Hom}_R(R/I, ...
3
votes
3answers
66 views

If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$, then $M$ has the structure of an $R/I$-module

Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: ...
-7
votes
1answer
133 views

Show that $T (S^{−1}M)= S^{−1}(T (M))$ [closed]

Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann_R(m)} \neq 0$, i.e. if $rm = 0$ for some nonzero $r \in R$. Denote the set ...
-4
votes
1answer
130 views

Show that $T(M)$ is a submodule of $M$.

Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann}_R(m) \neq 0$, i.e. if $rm =0$ for some nonzero $r\in R$. Denote the set of ...
0
votes
2answers
26 views

Does a module with the following properties necessarily finite length?

Let $M$ be a f.g. $R$-module, and suppose there exists a non-zero $r\in R$ such that $rM=0$. Does $M$ have finite length?
5
votes
2answers
182 views

$M\oplus A \cong A\oplus A$ implies $M\cong A$?

Let $A$ be a commutative unital ring and $M$ an $A$-module. Suppose that $M\oplus A \cong A\oplus A$. Then is $M\cong A$? We have that both $M\oplus A$ and $A\oplus A$ are biproduct for $(A, A)$ and ...
0
votes
0answers
51 views

The $I$-torsion submodules of an injective module [duplicate]

Possible Duplicate: Prove that the following module is injective Prove that $I$-torsion submodules of injective modules are injective (the ring is Noetherian). Please help me.
1
vote
0answers
78 views

ISBN Code - Construction, check digit example

I am doing a homework example and would like to ask your opinion on the correctness of the example: How is the ISBN 3-519-42227-1 constructed, at what point is the check digit and how to calculate ...
1
vote
1answer
89 views

Every subgroup of finite rank of $\mathbb Z^\mathbb N$ is free abelian

I want to prove that every subgroup of finite rank of $\mathbb Z^\mathbb N$ is free abelian. A group $G$ has finite rank $N$ if there exists a subset $ \{e_1,..,e_N\}$ of $G$ such that: $(1)$ If $a_i ...
2
votes
1answer
63 views

Same left KG-module induced by nonisomorphic G-sets

For any field $K$, a finite group $G$, and left $G$-set $X$, $KX$ has a obvious left $KG-$ module structure, where $KG$ is the usual group algebra generated by the group $G$. The question is Find ...
0
votes
1answer
78 views

Module Homomorphisms and a Direct Sum

I am trying to show the following: Let $f:M\to N$ and $g:N\to M$ be module homomorphisms such that $g\circ f=Id_M$. Prove that $N=Im(f)\oplus\ker(g)$. I know that $M/\ker(f)\cong Im(f)$, but I'm not ...
0
votes
1answer
44 views

What is a good way to prove a module is finitely generated if and only if the union of every chain of proper submodules of $M$ is a proper submodule?

The take-home exam give me the following problem: Prove that a module is finitely generated if and only if the union of every chain of proper submodules of $M$ is a proper submodule. For the if ...
-1
votes
1answer
99 views

Noetherian prime ideal is $R$-module and $R$ is Noetherian ring

If $R$ is a commutative ring, and for every $P$ a prime ideal of $R$, $P$ is a Noetherian $R$-module, show that $R$ is Noetherian.
1
vote
0answers
150 views

Hungerford Algebra - Chapter IV - proof of the splitting lemma

Im working with the splitting lemma right now and I don't understand it too much. If someone has that book and could help me complete the sketch in page 177, proof of 1.18) I would be very grateful. ...
1
vote
0answers
49 views

Counterexample for ${\rm Hom}_{R}(\prod_{i\in I}M_{i},N)\cong_{\mathbb{Z}} \coprod_{i\in I}{\rm Hom}_{R}(M_{i},N)$.

Let $\{M_{i}\}_{i\in I}$ be a family of $R$-modules and also $N$ is a $R$-module. Is there an counterexample for the following relation: ${\rm Hom}_{R}(\prod_{i\in I}M_{i},N)\cong_{\mathbb{Z}} ...
2
votes
1answer
70 views

calculating a quotient group of $\mathbb{Z}$-modules

I'm calculating a quotient group $A/B$ where $$A:=\left\{ ax+by+\sum c_i z_i \middle|\, a,b,c_i\in\mathbb{Z}, \sum c_i =0 \right\}$$ $$B:=\left\{ x\sum a_i +y\sum b_i +\sum (a_{i-1}-a_i + b_{i-1}-b_i) ...
4
votes
0answers
55 views

Character of $\text{Hom}_{\mathbb{C}}(\mathbb{C}G,\mathbb{C})$

I've been given the following two questions, and for both I'm really unsure what to do (I'm a beginner with the theory of characters). Let $G$ be a finite group, $\mathbb{C}G$ its group algebra over ...
2
votes
1answer
73 views

Confusion regarding what kind of isomorphism is intended.

For a class I'm taking this semester, I was given this question: (To guarantee clarity, I am quoting the full question even though my own question is only regarding the final sentence.) Let $G$ be ...
4
votes
1answer
95 views

Idempotent and Hermitian vectors in Group Algebra

Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
0
votes
1answer
40 views

Find $n$ value satisfying equations in $\mathbb Z_n$

Determine for which $n \in \mathbb N^*$ values the following equations in $\mathbb Z_n$ are satisfied: $[3] \cdot [5] = [3] + [5]$ $[3] \cdot [5] = [27]$ and $[3] + [5] = [11]$ I can't come up ...
1
vote
0answers
47 views

Free module, $\mathbb{Z}[a]$ over $\mathbb{Z}[(a+1)^2]$ for transcendental number a

I'm trying to prove that for a transcendental number $a$ the module $\mathbb{Z}[a]$ over $\mathbb{Z}[(a+1)^2]$ is free. For $\mathbb{Z}[a+1]$ over $\mathbb{Z}[(a+1)^2]$, the basis is $\{1,a+1\}$. What ...
6
votes
1answer
67 views

r-th transvectants and $\mathbb{C}G$-module maps

Suppose $V=\mathbb{C}^2$ and $G=SL(V)=SL_2(\mathbb{C})$. We define $C_n = H_{\mathbb{C},n}(V,\mathbb{C}) \cong S^n(V^*)$, the n-th symmetric power of the dual of $V$, i.e. the homogeneous polynomials ...
5
votes
1answer
118 views

Does similarity of integer matrices imply the transition matrix is an integer matrix?

I'm working on a homework question, and I'm stuck. The question is: Let $A$ and $B$ be $2n \times 2n$ rational matrices with $A^2=B^2=-Id$. The first part of the question asks to show that $A$ and ...
2
votes
1answer
145 views

A question with an odd hypothesis.

Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the ...
2
votes
3answers
180 views

$A$ is integrally closed if and only if $A_P$ is integrally closed for all maximal ideals $P$ of $A$.

I've managed [with help from the wonderful people lurking on this site :) ] to prove that for an integral domain $A$, if $A$ is integrally closed, then $S^{-1}A$ is integrally closed for all ...
2
votes
1answer
82 views

Interpreting an $S^{-1}R$-module as an $R$-module.

Can one do it? I'm trying to prove that $S^{-1}I$ is an injective $S^{-1}R$-module whenever $I$ is an injective $R$-module. So I need to start with a situation where I have: (i) $S^{-1}R$-modules ...
0
votes
0answers
46 views

$f,g$ $R$-module morphisms. Is it true that $gf$ is essential if and only if both $f$ and $g$ are essential?

Definition Let $f$ be a right $R$ module monomorphism. Say $f:A\to B$. We say that $f$ is essential if its image $f(A)$ is essential in $B$, that is, for any $C\leq B$ non zero, $f(A)\cap C\neq 0$. ...
0
votes
1answer
68 views

Equivalent characterization of essential (large) submodule

I'm stuck with the following question in the sense that I admit I cannot come up with a decent solution. Here is the question: We define a submodule $N$ of the $R$ module $M_R$ to be essential ...
9
votes
1answer
480 views

What does a zero tensor product imply?

I'm trying to prove that for two finitely generated $A$-modules $M,N$ ($A$ being any ring), the tensor product $M\otimes_A N$ is zero iff $Ann(M)+Ann(N)=A$. The if direction is of course easy- just ...
2
votes
1answer
109 views

How can we write this $\mathbb{Q}[x]$-module as a direct sum of cyclic $\mathbb{Q}[x]$-modules?

If $L$ is the submodule of $\mathbb{Q}[x]^{(3)}$ generated by $(2x-1,x,x),(x,x,x),(x+1,2x,x)$. How do we write $\mathbb{Q}[x]^{(3)}/L$ as a direct sum of cyclic modules?
10
votes
2answers
694 views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
0
votes
1answer
57 views

Prove a property of when simple modules are isomorphic

Let $\mathfrak{m}_1$ and $\mathfrak{m}_2$ be left maximal ideals of a unital ring $A$. Show that the simple modules $A/\mathfrak{m}_1$ and$A/\mathfrak{m}_2$ are isomorphic if and only if there exist ...
3
votes
2answers
225 views

For which prime $p$ are the additive groups $\mathbb{F}_p$ and $\mathbb{F}_p^2$ $\mathbb{Z}[i]$-modules?

Homework for my algebra class. Chapter 14, Exercise 7.8 in Artin's Algebra, Second Edition: Let $F = \mathbb{F}_p$. For which prime integers $p$ does the additive group $F^1$ have a structure ...
4
votes
1answer
284 views

A question about tensor products

This is about problem $5$ in Section IV.$5$ of Hungerford's Algebra book. The question is the following: If $A'$ is a submodule of the right $R$-module $A$ and $B'$ is a submodule of ...