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

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1answer
26 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
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3answers
31 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
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1answer
45 views

Tensor power modulo cyclic group action

Let $M$ be some $R$-module and $n \geq 1$ be some positive integer. The cyclic group $\mathbb{Z}/n\mathbb{Z}$, with a chosen generator $t$, acts on $M^{\otimes n}$ via $t(m_1 \otimes \dotsc \otimes ...
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1answer
30 views

If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right ...
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2answers
54 views

What is the significance of $A+ (B\cap C)=(A + B)\cap C$, where $A\subseteq C$, for modules?

My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here. Isn't ...
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0answers
56 views

Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
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0answers
58 views

Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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0answers
46 views

proving $\mathbb {Z}_n $ is an injective $\mathbb {Z}_n$-module [duplicate]

How can I prove a well-known fact: $\mathbb {Z}_n (=:\mathbb {Z}/n\mathbb {Z})$ is an injective $\mathbb {Z}_n $-module? I classified all ideals of $\mathbb {Z}_n $ and tried to use Baer's ...
4
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2answers
61 views

Flat Non Projective $A$-Module [duplicate]

A standard fact in Commutative Algebra is that a Projective $A$-module is flat. The converse is false. Can someone show me an example of a Flat Non Projective $A$-Module? Thank you!
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0answers
37 views

Yetter-Drinfeld modules over Hopf algebra

Let $H$ be a bialgebra and assume that $M$ is left $H$-module and left $H$-comodule. $M$ is called Yetter-Drinfeld module iff $\forall h\in H, \ \forall m \in M \ \ ...
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1answer
26 views

What is a composition series for $N$ and $M/N$

Let $M$ be a module with a composition series. Show that every submodule of $M$ is a member of a composition series for $M$. I understand the concept of a composition series for a module: it's a ...
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2answers
43 views

For an exact sequence $0\to M_1\overset{f_1}\to\cdots\overset{f_r}\to M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$?

For an exact sequence $0\to M_1\overset{f_1}\rightarrow\cdots\overset{f_r}\rightarrow M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$? $M_i$s are modules and $l(M_i)$ ...
4
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1answer
63 views

counterexamples on modules

Prove or disprove: for any comm. ring $R $, For three $R $-modules $M $, $N $ and $P $, $M \oplus P \simeq N \oplus P $ implies $M \simeq N $. Let $f:M\to N $ and $g:N\to M $ be two $R $-module ...
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1answer
62 views

$n+1$ elements in the free module $R^n$ must be linear dependent?

Suppose that $R$ is a commutative ring with unit and $R^n$ is the free module of dimension $n$ over $R$, it is right that $n+1$ elements in $R^n$ must be linear dependent?
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2answers
63 views

Example for an ideal which is not flat (and explicit witness for this fact)

I'm looking for an ideal $\mathfrak{a}$ of an commutative (possibly nice) ring $A$ together with an injective $A$-module homomorphism $M\hookrightarrow N$ such that the induced map ...
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1answer
41 views

Finite (cardinality) modules over a PID

Let $R$ be a principal ideal domain with $|R|=\infty$. Suppose $M$ is an $R$-module such that $2 \le |M| < \infty$. What properties does this imply about $R$? Background: I was hoping that $R\cong ...
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2answers
41 views

Are all injective endomorphisms of a module automorphisms?

As far as I understand, an automorphism is an isomorphism from a set to itself. If we have a homomorphism $f:M\rightarrow M$, then, from the first isomorphism theoreom, $im(f)$ is a submodule of $M$. ...
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1answer
43 views

Why are free modules called “free”?

Let $R$ be a ring (not necessarily commutative) with multiplicative identity. A $R$-module $M$ is called free if $M$ has a linearly independent generating set $\beta\subseteq M$. That is, for any ...
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1answer
320 views

Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
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0answers
43 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
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0answers
117 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
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0answers
54 views

Definition of tensor product using pushout.

Let $M, N$ be right and left $A$-modules respectively. Then we have actions $$ \varphi_M: M \times A \to M, \\ \varphi_N: A \times N \to N. $$ There is a definition of $M \otimes_A N$ as follows. We ...
5
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1answer
51 views

Uncountable injective submodule of quotient module of product of free modules

Let $I$ be uncountable set. How to prove that $\left(\prod\limits_{i\in I}\mathbb{Z}x_i\right)/\sum\limits_{i\in I }\mathbb{Z}x_i$ always contain uncountable injective module ? Can we construct it ...
2
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1answer
72 views

Countable submodule of free module

Is it true that if $F_0$ is a countable submodule of free $\mathbb{Z}$-module $F$ then $F/F_0$ doesn't contain any uncountable injective submodule ?
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1answer
29 views

Module of homomorphisms as injective module

We know that for any $R$-module exist injective $R$-module $\overline{M}$ such that there is inclusion $i:M\rightarrow \overline{M}$, where we treat $M,\overline{M}$ as $\mathbb{Z}$-modules. Show ...
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2answers
38 views

Equivalent condition for split exact sequence

the question is in Module Theory, Let $M,N,\&\ L$ be any R-modules. Then , for any short exact sequence $0\longrightarrow N\overset{f}{\longrightarrow} M\overset{g}\longrightarrow L ...
1
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1answer
57 views

Tensor product of modules and being torsion free

$(i\otimes_{\mathbb{Z}} 1_N):M'\otimes_{\mathbb{Z}} N \rightarrow M\otimes_{\mathbb{Z}} N$ is a monomorphism for every monomorphisms $i:M'\rightarrow M$ iff $\mathbb{Z}$-module $N$ is ...
4
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1answer
100 views

Properties of tensor product of modules

Let $M'$ be a submodule of $\mathbb{Z}$-module $M$, and let $i:M'\rightarrow M$ be a natural monomorphism. How to prove the following theorem ? : $i\otimes 1_N:M'\otimes_{\mathbb{Z}} N ...
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2answers
69 views

Isomophisms of modules

I'm reading a book about homological algebra. There is one exercise with whom I have a problem. Show that for any $\mathbb{Z}$-module $M$ and any $q\in \mathbb{Z}$ we have I) $ \ \ ...
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0answers
54 views

Is there an example such that $\text{rank}(A^t)\neq \text{rank}(A)$?

Let $R$ be a commutative Noetherian ring and $A\in M_{n\times m}(R)$. If $R$ is a field, then $rank(A^t)=rank(A)$. However, in general Noetherian rings, does $rank(A^t)=rank(A)$ hold? Since the ...
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0answers
31 views

Is it true that $M\otimes_A F\simeq M^{(I)}$?

Let $A$ be an $R$-algebra ($R$ is a commutative ring with identity $1_R$) and suppose $F$ is a left free module over $A$. Is it true that $$M\otimes_A F\simeq M^{(I)}$$ for any right module $M$ over ...
2
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0answers
40 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
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1answer
35 views

Non-zero maps between modules

Let $R$ be a ring and $M,N$ two $R$-modules such that there exists a non-zero map $\psi : M \to N$. Is it true that there exists a non-zero map $\varphi: N \to M$ ? I do not have any particular ...
0
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1answer
21 views

Exact sequences of projective modules

Let $0{\rightarrow} K \stackrel{g}\rightarrow P\stackrel{f}\rightarrow Q \rightarrow 0$ and $0\rightarrow K' \stackrel{g'}\rightarrow P'\stackrel{f'}\rightarrow Q \rightarrow 0$ be exact sequences ...
2
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1answer
57 views

Ideals and injective modules

Let $I$ be a left ideal of $R$. Assume that there exist element in $I$, which is not a zero divisor. How to prove that for every (left) injective $R$-module $Q$ we have $IQ=Q$ ? I need only hints. ...
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2answers
29 views

Injective modules

Let $\{R_n\}_{n\in \mathbb{N}}$ be a collection of fields. Let $R:=\prod\limits_{n=1}^{\infty}R_n$. Show that for every $n$ $R_n$ is an injective $R$-module, but $\bigoplus\limits_{n=1}^{\infty}R_n$ ...
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1answer
40 views

Prove that there's a unique morphism that completes the commutative diagram

I have to prove that there's a unique $\gamma : M'' \rightarrow N''$ that completes this diagram considering the rows are exact. $$\begin{array} MM' \stackrel{f_1}{\longrightarrow} & M & ...
2
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1answer
31 views

Another approach to $A/I\otimes_A A/J\simeq A/(I+J)$?

The same question appears here $A/ I \otimes_A A/J \cong A/(I+J)$ however I'm looking for a different approach. Let $A$ be an algebra, $I$ a right ideal and $J$ a left ideal. I'd like to show ...
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1answer
34 views

Ideal as projective module

Let $R$ be a commutative ring without zero divisors. Assume that ideal $a\subset R$ is a projective $R$-module. How to prove that $a$ is finitely generated ? I need only hints.
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1answer
74 views

When is $M\otimes N$ a module?

So suppose we have a $R$-$S$-bimodule $M$, and an $S$-$T$ bimodule $N$, then we can construct the abelian group $M\otimes_S N$. Under what conditions could we make this a module? I would expect this ...
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1answer
18 views

Ring of polynomials as free module

Is it true that $R=k[x,y]$ is a free $R$-module ? I think that it isn't true. Natural candidate for the base is $\{x^{\alpha}y^{\beta}\}_{\alpha,\beta}$, but : $x\cdot (xy) +(-y)\cdot x^2 =0$ and ...
4
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0answers
55 views

Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
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0answers
35 views

Looking for reference to solve a problem (representation theory, I think)

This question appeared on an algebra qualifying exam: Let $R$ be the group algebra $\mathbf{C}[S_3]$. How many nonisomorphic, irreducible, left modules does $R$ have and why? Would I be able to ...
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1answer
31 views

Find all the possible pairs of submodules $M_1$ and $M_2$ of the $\mathbb{Z}$-module $\mathbb{Z}_{18}$ so that $\mathbb{Z}_{18} = M_1 \oplus M_2$

I started by considering the possible proper subgroups of $\mathbb{Z}_{18}$, which are $\langle\bar{9}\rangle$, $\langle\bar6\rangle$, $\langle\bar3\rangle$ and $\langle\bar2\rangle$, which are also ...
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1answer
21 views

Radical of the annihilator of an element of a Noetherian module

Assume $M$ is a commutative Noetherian $R$-module and $m\in M$ is such that $P=\sqrt{\operatorname{Ann}(m)}$ is a prime ideal in $R$. Is it true that $P$ is an associated prime of $M$, i.e. there is ...
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1answer
13 views

About extensions of finitely generated modules.

Let $R$ be a ring with unity, and $A \subseteq B$ be $R$-modules. I want to know if the following is true: If $A, B/A$ are finitely generated $R$-modules, then $B$ is finitely generated. Under ...
2
votes
2answers
39 views

Characterizing the A-module M/S

I've been working through this for a little while, and I'm not 100% sure I understand what I'm supposed to be doing here, or maybe I'm not grasping correctly what they mean by "Characterize". ...
4
votes
1answer
78 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
2
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1answer
81 views

$\operatorname{Hom}_R(\mathfrak{a},M)$ is isomorphic to $\mathfrak{a}^{-1}M$ if $R$ is a Dedekind domain

I want to prove Lemma 2.5.1 of Silverman's Advanced Topics in The Arithmetic of Elliptic Curves (whose proof is left to the reader): Let $R$ be a Dedekind domain, let $\mathfrak{a}$ be a ...
1
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1answer
52 views

What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...