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

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3
votes
1answer
143 views

If $x\otimes y=0$, then $x\otimes y=0$ in the tensor product of finitely generated submodules

Let $A$ be a commutative ring, let $E,F$ be $A$-modules. Let $x\in E$ and $y\in F$. If $x\otimes y=0$ in $E\otimes F$, then there exist finitely generated submodules $M\subset E$ containing $x$ and ...
6
votes
2answers
358 views

Equivalent definitions for projective modules

Fact: Let $R$ be a ring with identity. Let $J$ be an $R$-module. Then, $J$ is injective iff for every left ideal $L$ of $R$ every $R$-module homomorphism $L\rightarrow J$ can be extended to an ...
2
votes
1answer
129 views

When is a quotient of an $R$-module $E$ a submodule of $E$?

Let $R$ be a commutative ring with $1$. Suppose we are given a surjective $R$-module map $\varphi:E \to M \simeq E/N$. Are there any sufficient and/or necessary conditions for having an injective map ...
0
votes
1answer
48 views

Why $(M/M \operatorname{rad} A) \operatorname{rad}A=0$?

Let $A$ be a ring and $M$ a right $A$-module. Why we have $(M/M \operatorname{rad}A) \operatorname{rad}A=0$? Thank you very much.
3
votes
1answer
55 views

How to show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?

Let $M, N$ be right $A$-modules. How can we show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?
3
votes
1answer
105 views

Question about radical of a module.

Let $M$ be a right $A$-module. How to show that $m\in \operatorname{rad}(M)$ iff for any simple right $A$-module $S$ and any $f\in \operatorname{Hom}_A(M, S)$, $f(m)=0$? I think that if $m$ is ...
3
votes
1answer
158 views

Injectivity is a local property

Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How can I show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$, then $M$ is ...
5
votes
1answer
119 views

$M$ is $\bigcap \operatorname{Ass}(M)$-primary

Let $R$ be noetherian ring and $M$ an $R$-module such that $\operatorname{Ass}(M)$ is a finite set. Prove that $M$ is $\mathfrak{b}$-primary, where $\mathfrak{b}=\bigcap ...
1
vote
1answer
26 views

If $u$ and $v$ have the same order, $u=f(\tau)(v)$ iff $v=g(\tau)(u)$?

Suppose $\tau\in L(V)$, for $V$ a finite dimensional vector space, and $p(x)$ is an irreducible factor of the minimal polynomial $m_\tau(x)$. Also, suppose $u,v\in V$, and both $u$ and $v$ has order ...
9
votes
1answer
883 views

Why isn't an infinite direct product of copies of $\Bbb Z$ a free module?

Why isn't an infinite direct product of copies of $\Bbb Z$ a free module? Actually I was asked to show that it's not projective, but as $\Bbb{Z}$ is a PID, so it suffices to show it's not free. ...
5
votes
1answer
141 views

If $V$ is a vector space over $k$, is every $k[x]$-module structure on $V$ induced by some linear transformation?

Suppose $V$ is a vector space over a field $k$. Fixing a linear transformation $T$, it is common to make $V$ a $k[x]$-module by defining $f(x)\cdot v=f(T)(v)$. Is every possible $k[x]$-module ...
3
votes
1answer
164 views

Free module such that $R=R\oplus R$

I was wondering if the following is correct: Let $V=\bigoplus_{i\in\mathbb N}\mathbb Z$ and $R=\text{Hom}_{\ \mathbb Z}(V,V)$. Regard $R$ as an $R$-module. Then $R$ is free of rank $1$ with basis ...
3
votes
2answers
111 views

Maximal ideals generate maximal submodules?

Let $\mathfrak m$ be a maximal ideal of $R$ and $M$ an $R$-module such that $\mathfrak mM\ne M$. Is it true that $\mathfrak mM$ is a maximal submodule of $M$? Thank you. (I can see this happen in ...
11
votes
2answers
138 views

How many ways are there to consider $\Bbb Q$ as an $\Bbb R$-module?

How many ways are there to consider $\Bbb Q$ as an $\Bbb R$-module? I guess there is only one way, and that is the trivial case. i.e. $$\forall r\in \Bbb R,\, \forall a,b\in \Bbb Q \qquad r\cdot ...
0
votes
1answer
38 views

Modular Inverse and Chinese Remaindering

I am given a positive integer $n$, two elements $\alpha, \beta \in \mathbb{Z}^*_n$, and integers $l$ and $m$, such that $\alpha^l = \beta^m$ and $\gcd(l, m) = 1$. How do I compute $y \in ...
2
votes
1answer
98 views

When do finiteness conditions coincide on modules?

Let $R$ be a (possibly noncommutative, but unital) ring. There are several finiteness conditions on a left $R$-module that we can consider. The ones I have come across include finitely generated ...
1
vote
1answer
212 views

Is there a unique cyclic submodule for every factor dividing the order of a cyclic module?

Suppose $\langle v\rangle$ is a cyclic module over a PID of order $a$. Then there is a cyclic submodule of order $b$ for any $b$ dividing $a$. Namely, $\langle cv\rangle$, where $c$ is such that ...
3
votes
1answer
72 views

Module Question: Defintion Basis and Free

I have been working with modules for some time now, but there are a two little things I have been wondering about: (1) defintion of a basis (2) defintion of a free module. They may seem trivial to ...
1
vote
1answer
32 views

Free part of a module decomposition is not necessarily unique.

I am reading some algebra notes (http://www.math.umn.edu/~garrett/m/algebra/notes/11.pdf). Corollary 2.0.6 says that if $R$ is an integral domain, any finitely generated module module $M$ is ...
2
votes
3answers
104 views

If $\operatorname{rk}(N)=1$, and $M/N$ is torsion, why is $\operatorname{rk}(M)=1$?

Suppose $M$ is a finitely generated torsion-free module over a PID. If $N\leq M$ is free of rank $1$, and $M/N$ is torsion, how do we conclude $M$ is free of rank $1$? My scattered thoughts: Since ...
2
votes
1answer
117 views

Why is the cyclic decomposition of a primary torsion module not unique?

It is known that a finitely generated torsion module $M$ over a principal ideal domain $R$ can be decomposed into a direct sum of primary modules, $$ M=M_{p_1}\oplus\cdots\oplus M_{p_n}. $$ ...
3
votes
1answer
124 views

Finitely generated modules over a principal ideal domain.

Let $M$ be a module, finitely generated,over the principal ideal domain $D$. If $D$ is a field, then $M$ has a basis. But if $D$ is not a field theis is not truth. Can you give me an example of a ...
6
votes
2answers
385 views

Tensor product over different rings are not equal

I'm looking for an example of modules $A,B$ such that $A\otimes_{R}B\neq A\otimes_{\mathbb{Z}}B$. I've thought about some tensor products I know but I haven't been able to produce an example. What ...
3
votes
2answers
203 views

Coproducts and products of modules

I've just looked at the book "A Course in Homological Algebra" ( by Hilton and Stammbach) . They show the universal property of the direct sum (coproducts) using injections and the universal property ...
3
votes
1answer
496 views

Tensor product of module homomorphisms

This is an exercise problem from Hungerford's Algebra but first I'll state a result that forms the background to the problem. Let $R$ be a ring. Let $A,A'$ be right $R$-modules. Let $B,B'$ be left ...
2
votes
2answers
78 views

Is this function from $M_g \to \mathbb{Z} \otimes M$ well defined?

Given a $G$-module $M$ (a $\mathbb{Z}G$-module), let $M_g$ denote the group of coinvariants, that is $M/\langle m -gm \rangle$. If we regard $\mathbb{Z}$ as a right $\mathbb{Z}G$-module with trivial ...
0
votes
1answer
48 views

Question on modules.

Let $\mathbb Z \subset A$ be integral domains. If $A$ is a finite $\mathbb Z$-module then how to show that there exist $w_1,w_2,\dots,w_n \in K$ such that $A$ is isomorphic to the subring $\mathbb ...
11
votes
7answers
2k views

Applications of the Isomorphism theorems

In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I ...
4
votes
1answer
143 views

quotient of finitely presented module

Assume the exact sequence $$ \begin{equation} A \xrightarrow{f} B \xrightarrow{} Cokerf \xrightarrow{} o \end{equation} $$ where $A$ is a finitely generated module and $B$ is a finitely presented ...
2
votes
1answer
282 views

When is the localization of a module trivial?

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. Let $S^{-1}M$ be localization of $M$, where $S$ is a multiplicatively closed subset of $R$. How to show that $S^{−1}M =0$ if ...
2
votes
1answer
50 views

Question on decomposition of cyclic modules.

I have a question on the following proof, which takes place over an $R$-module $M$ where $R$ is a PID. Here $v\in M$, and $o(v)$ is the order of $v$, defined to be a generator (or any of its ...
1
vote
0answers
64 views

Cancellation property for modules over nonabelian groups

Let $\Gamma$ be a nonabelian group, and let $M_1, M_2$ be modules over $\mathbb{Z}\Gamma$ (let us say finitely generated). Suppose that $M_1 \oplus\mathbb{Z}\Gamma=M_2\oplus\mathbb{Z}\Gamma$. Does it ...
3
votes
1answer
56 views

Examples of modules such that $M\oplus N_1\simeq M\oplus N_2$, but $N_1\not\simeq N_2$? [duplicate]

Are there any examples of modules such that $M\oplus N_1\simeq M\oplus N_2$, but $N_1\not\simeq N_2$ as modules? I thought of taking $\bigoplus_{i\geq ...
6
votes
1answer
1k views

Specific proof that any finitely generated $R$-module over a Noetherian ring is Noetherian.

I have seen a handful of proofs that any finitely generated module over a Noetherian ring is again Noetherian. I'm specifically trying to understand the following proof idea. It goes as this: Observe ...
2
votes
1answer
137 views

Injection from an integral domain to its field of fractions.

I have a quick question about modules. Suppose that $R$ is an integral domain with field of fractions $K$. Then any free $R$-module is isomorphic to copies of direct sums of $R$, say $R^i$ . ...
16
votes
1answer
204 views

Can extending a finite ground field make modules isomorphic?

$\def\Hom{\mathrm{Hom}}$Let $k$ be a field, $A$ a $k$-algebra and let $M$ and $N$ be $A$-modules, finite dimensional over $k$. Let $K$ be an extension of $k$, so $A \otimes K$ is a $K$-algebra and $M ...
1
vote
1answer
81 views

All associated primes appear in a series of submodules

This is essentially Ex VI.4.8 of Algebra: Chapter 0. Let $R$ be a commutative ring and $M$ be an $R$-module. Define \begin{equation} \operatorname{Ann}_{R}(m)=\{r\in R:rm=0\} \end{equation} for each ...
5
votes
2answers
882 views

Difference between free and finitely generated modules

I am not sure I understand the difference between free modules and finitely generated modules. I know that a free module is a module with a basis, and that a finitely generated module has a finite set ...
1
vote
1answer
2k views

The (Jacobson) radical of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. Let $\Omega$ be the set of all maximal ideals of $R$. Prove that $\operatorname{Rad}(M)=\bigcap_{\mathfrak m\in \Omega}\mathfrak mM$, where ...
0
votes
1answer
101 views

Isomorphic completed modules, that were not isomorphic before completion

Let $M$ and $N$ be $R$-modules. Suppose we complete them with respect to an ideal $\frak{m}$ of $R$. If we have $$M^\wedge_\mathfrak{m} \simeq N^\wedge_\mathfrak{m}$$must if be the case that $M ...
2
votes
3answers
79 views

Existence of a certain R-module homomorphism

Let $R$ be a ring with identity and $A,B,C$ be $R-$modules. Let $f:A\rightarrow C, g:B\rightarrow C$ be $R$-module homomorphisms such that $g$ is surjective. Does there exist an $R-$module ...
4
votes
3answers
294 views

Isomorphisms between tensor products field of fractions of integral domain

Let $D$ be an integral domain. How can I show that there are isomorphisms $$F(D) \otimes_D F(D) \to F(D) \otimes_{F(D)} F(D) \to F(D)$$ The identity map seems like the natural choice for the first ...
2
votes
2answers
68 views

What is an easier way to think about tensor products of modules?

I'm having trouble understanding the the tensor product of modules. Sure I get the "it's quotient of the free abelian group by the subgroup generated by certain elements" construction, but this ...
6
votes
2answers
1k views

Isomorphism between tensor product and quotient module.

Given a commutative ring $R$ with $1$, an ideal $I$ and a left $R$ module $A$, how can I show that the homomorphism $$f: R/I \otimes_R A \to A/IA$$ given by $f((r +I) \otimes a) = ra + IA$ is ...
2
votes
1answer
88 views

Free $R$-module $G$ with countably infinite basis isomorphic to direct sum of $G$ with $G$.

Let $G$ be a free $R$-module on a countably infinite set. How can I show that $G \cong G \oplus G$. This is clearly false for finite sets, so how does the fact that it's countably infinite make ...
1
vote
1answer
122 views

Total divisor in a Principal Ideal Domain.

Let $R$ be a right and left principal ideal domain. An element $a\in R$ is said to be a right divisor of $b\in R$ if there exists $x \in R$ such that $xa=b$ . And similarly define left divisor. $a$ ...
0
votes
2answers
519 views

Submodule of a semisimple module is semisimple .

I have a doubt in proving the above statement . Lets say that $M$ a semisimple module , then consider $P \subset N$ , $P$ and $N$ are two sub-modules of $M$ . now since $M$ is a semisimple module ...
0
votes
3answers
330 views

Computing Large Powers

So this is my question: Compute $7^{818} \pmod {1637}$ using no more than 14 multiplications mod 1637. (You should of course verify that 1637 is prime if you plan to use Fermat's Theorem.) I would ...
3
votes
1answer
200 views

vector bundles on the affine line over a PID

Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial? For $R=k[X]$ this is true by the Theorem of ...
4
votes
1answer
209 views

Simple module over a basic algebra is one-dimensional

May I refer you to page $32$ of: http://www.staff.city.ac.uk/a.g.cox/LTCC/Week3.pdf Part (b) of proposition $3.2.4$. Why the fact that $A/\operatorname{rad}(A)$ is isomorphic to $k^m$ for some $m$ ...