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

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2
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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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
8
votes
1answer
145 views

Infinite linear independent family in a finitely generated $A$-module

So I'm stuck with this problem. Let $A$ be a commutative ring (with unit). I have several questions that are really close to each other. 1) Let $M$ be a finitely generated module over $A$. Can we ...
1
vote
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 ...
0
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1answer
18 views

Projective space of a module

I'm studying projective geometry in a basic course of geometry. My question is: Is there an equivalent definition of projective space not of a vector space but of a module? I think the basic ...
8
votes
3answers
195 views

Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement ...
0
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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 ...
3
votes
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 ...
2
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1answer
55 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

Find a non-commutative ring with exactly 2014 two sided-proper ideals. Find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have ...
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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
votes
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 ...
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 = ...
3
votes
1answer
568 views

Finding a basis for a submodule

Let $F$ be the $\mathbb{Z}$ -module $\mathbb{Z}^{3}$ and let $N$ be the submodule generated by {$(4,-4,4),(-4,4,8),(16,20,40)$}. Find a basis $\left\{ f_{1},f_{2},f_{3}\right\}$ for $F\textrm{ ...
0
votes
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
votes
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.
0
votes
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
votes
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. ...
0
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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 $\bigoplus\limits_{n=1}^{\infty}R_n$ ...
1
vote
1answer
15 views

Can one extent to (left) modules on IBN ring the results of vector spaces?

We know that (left) modules "are not as nice as vector spaces" because lots of properties of these ones don't fit to the most general modules. But il we restrict to modules on IBN ring (i.e. modules ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
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". ...
0
votes
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
3answers
165 views

A simple example of a ring that is an $A$-module but not an $A$-algebra

Let $A$ and $B$ be commutative rings, and suppose (the underlying group of) $B$ has a structure of $A$-module. "Obviously", that doesn't imply that $B$ gets a structure as an $A$-algebra, but I can't ...
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 ...
1
vote
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 ...
1
vote
3answers
105 views

All simple modules are projective $\Rightarrow$ semisimple [duplicate]

Let $A$ be a finite dimensional algebra over a field $K$. It is clear that if $A$ is semisimple, then every simple module is projective. Does the converse hold ? It seems false, but I can't find a ...
5
votes
1answer
151 views

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
1
vote
1answer
39 views

$M_1$, $M_2$ and $M_1\cap M_2$ injective imply $M_1+M_2$ is also injective?

Let $M$ be an $R$-module ($R$ is a ring with identity) and let $M_1$ and $M_2$ be two injective submodules such that $M_1\cap M_2$ is also injective. How to show $M_1+M_2$ is injective? If the ...
0
votes
0answers
21 views

Fiber product of $f$ and $g$ is isomorphic to $\mathbb Z\oplus \mathbb Z_p$?

Let $p$ be a prime number. I'm supposed to show the fiber product (pullback) of the canonical projections $f:\mathbb Z\longrightarrow \mathbb Z_{p}$ and $g:\mathbb Z_{p^2}\longrightarrow \mathbb Z_p$ ...
5
votes
2answers
336 views

How to prove surjectivity part of Short Five Lemma for short exact sequences.

Suppose we have a homomorphism $\alpha, \beta, \gamma$ of short exact sequences: $$ \begin{matrix} 0 & \to & A & \xrightarrow{\psi} & B & \xrightarrow{\phi} & C & \to & ...
0
votes
1answer
16 views

module isomorphism inbetween two equivalence classes of polynomials

Let $g \in \mathbb{R}[t]$ be a normed irreducible polynomial of degree 2, meaning that $g(t) = (t - \lambda)(t - \overline{\lambda}$) for a $\lambda = a + b i$, with $a, b \in \mathbb{R}$, $b ≠ 0$. I ...
1
vote
1answer
37 views

How to define a group ring when the group is infinte?

I am trying to prove the following from Tadao Oda's "Convex bodies and algebraic geometry" Let $N \cong \mathbb Z^n$ be a free $\mathbb Z$ - module of rank $n$ and $M = \text{Hom}_{\mathbb ...
3
votes
1answer
76 views

global dimension of rings and projective (flat) dimension of modules

Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite? Similarly, Let $R$ be ring such that ...
3
votes
1answer
45 views

Difference between $(N+P)/N$ and $P/N$

If $N$ and $P$ are submodules of the $A$-module $M$ (where $A$ is a commutative ring with unity), why is there a difference between $(N+P)/N$ and $P/N$? If $x\in (N+P)/N$ then $x=n+p+N=p+N$ for some ...
0
votes
2answers
44 views

Prove that $M$ is finitely generated and it is a semi-simple module.

Let $M$ be a $R$-module. It is given that intersection of all maximal sub-modules of $M$ is the zero module. Moreover the module is given to be Artinian. Prove that $M$ is finitely generated ...
2
votes
3answers
41 views

Prove that if $M$ is finitely generated then it is Artinian.

Let $M $ be a semisimple $R$-module. Prove that if $M$ is finitely generated then it is Artinian. To show this we have to prove that every non-empty collection of sub-modules of $M$ has a minimal ...
4
votes
1answer
65 views

Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
2
votes
1answer
33 views

why is a simple ring not semisimple?

A simple module is a semisimple module . A module $M$ is called semisimple if every submodule is a direct summand of $M$ Since a simple module has $\{0\}$ and $M$ as its submodules so it is ...
6
votes
1answer
130 views

Prove $AB=I$ implies $BA=I$ using Fitting's Lemma

I know this question has already been asked, but I need a proof that for $A,B \in M_n(K)$, $$AB=I_n \Rightarrow BA=I_n$$ using Fitting's lemma. I thought of using the fact that $K^n$ is a ...
1
vote
2answers
22 views

Do modules have to be defined over rings with unity?

This is definition for left module over a ring $R$ given in Wikipedia: Suppose that $R$ is a ring and $1_R$ is its multiplicative identity. A left $R$-module $M$ consists of an abelian group ...
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1answer
25 views

A question from the proof of: If M is a free f.g R-module, and N is a submodule of M, then N is free, where R is a PID

I'm reading the proof from Dummit and Foote third edition, page 460-461,theorem 4. I am confused about one thing: On page 461, there are: (a) $M=Ry_1\oplus \ker v$ (b)$N=Ra_1y_1\oplus (\ker v\cap ...
3
votes
2answers
52 views

relation of R-module homomorphisms with direct sums

If $f$ is a module homomorphism from $A$ to $B$ and $g$ is a module homomorphism from $B$ to $A$, and $g \circ f$ is the identity function, why is it that $B= \operatorname{im}(f) + \ker(g)$?
4
votes
1answer
456 views

Trivial extension of an algebra

Suppose that $A$ is a finite dimensional $k$-algebra. Call $Q=\mathrm{Hom}_k(A,k)$. $Q$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial extension of $A$ is defined as follows: ...
2
votes
2answers
69 views

Characterization of projective modules?

I'm having a hard time with a characterization of projective modules: A $R$-module $P$ is projective if and only if for every epimorphism $f:I\longrightarrow I^{\prime\prime}$ with $I$ injective and ...
2
votes
1answer
25 views

Finitely generated and free module

We work in the category $R-\mathsf{Mod}$ where $R$ is unital. Some authors define free modules to have a finite basis. If we don't require a basis to be finite, I think it is quite obvious that a ...
0
votes
1answer
37 views

Every projective $R$-module $P$ is free

I have come across a theorem which states that if the underlying ring $R$ is a principal ideal domain then every $R$-module $P$ which is projective is free also. But the problem is I have encountered ...