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

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4
votes
1answer
58 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 ...
0
votes
0answers
26 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 ...
4
votes
0answers
49 views

Example of $A$-module but not $A$-algebra. [duplicate]

If $A$, $B$ are commutative rings, and if $B$ is an $A$-algebra then it is also an $A$-module. I am looking for an example that shows that the converse is not true. That is, I am looking for ...
1
vote
1answer
43 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
1answer
36 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 $g:\mathbb Z_{p^2}\longrightarrow \mathbb Z_p$ ...
1
vote
1answer
31 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 ...
0
votes
1answer
13 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 ...
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 ...
1
vote
2answers
35 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
37 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 ...
0
votes
0answers
26 views

Graphs associated with rings and modules

There are several articles in the literature that deals with some interesting graphs associated with rings and modules. For example The zero-divisor graphs D. F. Anderson, P. S. Livingston, The ...
3
votes
1answer
97 views
+50

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 ...
4
votes
1answer
54 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 ...
-1
votes
2answers
26 views

Showing that a specific surjective $R$-module homomorphism is an isomorphism [on hold]

Suppose that $M$ is a Noetherian $R$-module and $f$ is a surjective $R$-module homomorphism from $M$ to $M$. Is $f$ an isomorphism?
1
vote
2answers
20 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 ...
1
vote
1answer
24 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 ...
0
votes
0answers
8 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 ...
3
votes
2answers
48 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 $B= \operatorname{im}(f) + \ker(g)$?
2
votes
1answer
23 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
32 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 ...
0
votes
0answers
38 views

How bad must be a ring to allow cyclic artinian modules that are not noetherian?

I've been studying the relations between artinian and noetherian modules over commutative rings. One can prove two interesting results for the commutative case. Theorem Every commutative artinian ...
1
vote
0answers
33 views

Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
0
votes
1answer
25 views

Inverting a nonzerodivisor of a module

I'm reading the Paper "What makes a complex exact?" by Eisenbud and Buchsbaum. On page 266 it says: Thus we may assume $0 \neq \operatorname{rank}(\phi_n,L) < \operatorname{rank}(F_n)$ and ...
-1
votes
3answers
34 views

Is there an easy example of a module which is not projective

Is there an easy example of a module which is not projective? I found that direct product of $\mathbb Z$ is not projective but its proof is complicated. Are there any easy examples for this?
2
votes
2answers
59 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
0answers
21 views

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times…$ is not projective as a $\mathbb Z$ module? [duplicate]

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times...$ is not projective as a $\mathbb Z$ module? I know that $\mathbb Z$ is a free $\mathbb Z$ module since it has ...
5
votes
0answers
23 views

Why not take the tensor product of two left modules in this way? [duplicate]

Let $A,B$ be two left $R$-modules. I was wondering if we then can form the tensor product of $A$ and $B$ by the free abelian group on $A \times B$ divided out by the span of the following elements ...
-2
votes
1answer
24 views

Quotient module and submodule.

Set $M$ as an $R$-module, where $R$ is a PID, then if any quotient module of M can be seen as a submodule of M? It seems not always true, and could you give me a counterexample? Thank you!
0
votes
1answer
30 views

If $M=\langle v_1,…,v_n\rangle $ is an $R$-module, is $\langle v_1\rangle$ always a submodule of $M$?

Note $R$ is a PID. In my mind, as $\langle v_1\rangle$ is a subgroup of $M$, but maybe it's not closed under scalar multiplication. Could you give me an counterexample, if $\langle v_1\rangle $ is ...
0
votes
0answers
50 views

What are the ring theoretic properties of this trivial extension? [closed]

Let $R={\mathbb Z }_{ (p) }{ \ltimes\mathbb Z }_{ { p }^{ \infty } }$ be the trivial extension of ${\mathbb Z }_{ (p) }=\{ a/b\in\mathbb Q:b\neq 0,\gcd(a,b)=1\text{ and } p\nmid b\}$ by ${\mathbb Z ...
0
votes
2answers
44 views

Proof about finitely generated torsion-free R-module M is free, where R is a PID

Proof about finitely generated torsion-free $R$-module $M$ is free, where $R$ is a PID. Can I prove by the following way? Proof by induction on $n$, where $M=\langle v_1,...,v_n\rangle$. If ...
0
votes
0answers
17 views

Finite submodules of infinite torsion module

Let $R$ be a commutative ring and let $M$ be an infinite torsion $R$-module. Does $M$ have an infinite sequence of finite submodules of increasing cardinality? If not, is it true given additional ...
1
vote
3answers
96 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 ...
1
vote
0answers
16 views

Computing injective hulls over a lower triangular matrix ring

Let $R$ be the ring $\begin{pmatrix} {\mathbb Z }_{ p } & 0 \\ {\mathbb Z }_{ p } & {\mathbb Z }_{ p } \end{pmatrix}$ with $p$ prime. $R$ is an artinian ring and is also a $\mathbb ...
6
votes
1answer
126 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 ...
4
votes
1answer
51 views

Right modules Vs Left modules.

I have been reading Frobenius Algebras, Volume 1 By Andrzej Skowroński, Kunio Yamagata. On page 18 I came across the following paragraph, and I founded interesting, I will quote it and then ask my ...
1
vote
2answers
21 views

Proving that an $FG$-homomorphism is surjective

Assume that $V$ is an $FG$-module.Prove that the subset $$V_0 = \{v \in V : vg = v \space \forall \space g \in G \}$$ is an $FG$- submodule of $V$. Also show that the function $$\phi: v \to ...
-2
votes
2answers
41 views

Is every epimorphism $M\longrightarrow A$ a retraction? [closed]

Let $M$ be a $A$-module (where $A$ is an algebra over a commutative ring $R$). How can I show: $(i)$ Every epimorphism $M\longrightarrow A$ is a retraction; $(ii)$ There are monomorphisms ...
1
vote
0answers
36 views

is $Hom(P,N \otimes_{End(P)} P) = N$?

This is probably well known to people who work with algebras but I couldn't find a reference. Say I have a ring A and a module P and I take B = End(P), the endomorphism ring. Let N be a B-module, is ...
0
votes
1answer
17 views

Diagram with exact sequences of modules

I am trying to solve these two problems on diagrams of module morphisms: 1) Let $$\begin{array}{c} M' & \xrightarrow{f_1} & M & \xrightarrow{f_2} & M'' \\ & & ...
4
votes
0answers
47 views

Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
0
votes
1answer
29 views

Elements of $M\otimes_{\mathbb K}\mathbb K[t]$?

I'm studying tensor product of modules over algebras and I came across the following statement. Given a $\mathbb K$-module $M$, the elements of $M\otimes_{\mathbb K}\mathbb K[t]$ can be written as ...
0
votes
1answer
6 views

Show that any two elements of $M$ are linearly dependent over $\mathbb Z$.

Let $M$ be the additive group of rationals .Show that any two elements of $M$ are linearly dependent over $\mathbb Z$. I do not understand why is this true.If $c_1\dfrac{a}{b}+c_2\dfrac{c}{d}=0$ then ...
2
votes
0answers
30 views

homology commutes with direct product of chain complexes. Direct proof

This is an attempt to prove that direct product of chain complexes commutes with homology (exercise in Weibel's book). I've had some success since I've proved that $Z_n(\prod_{\alpha \in A} C_{\alpha ...
0
votes
1answer
20 views

Weibel exercise 1.1.2. the $n$th homology module is a functor from category Ch-Mod$(R)$ to Mod-$R$

Ch-Mod$(R)$ is the category of $R$-module chain complexes. How do you turn a homology module into a functor? Thanks for teaching.
0
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0answers
16 views

Weibel's book exercise 1.1.2. Cycles get sent to cycles by chain complex homs $u : C_{\cdot} \to D_{\cdot}$

A morphism of chain complexes is a family of homs $u_n : C_n \to D_n $ such that $u_{n-1} d_n^{(C)} = d_n^{(D)} u_n$. Weibel's book says that cycles "get sent to cycles". To me that means that ...
1
vote
1answer
32 views

Is this a typo in Weibel, page 1?

It says a morphism $u : C_{\cdot } \to D_{\cdot}$ of chain complexes is a family of homomorphisms $u_n : C_n \to D_n$ such that $u_{n-1} d_n = d_{n-1} u_{n}$, but shouldn't it just be that $u_{n-1} ...
0
votes
0answers
25 views

How do I write a correct answer to Weibel exercise 1.1.1.?

Exercise 1.1.1. Set $C_n = \Bbb{Z}/8$ for $n \geq 0$ and $C_n = 0$ for $n \lt 0$. Let $d_n : x \pmod{8} \to 4x \pmod{8}$ Compute the homology modules of the chain complex $C_{\cdot}$. I got that ...