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

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3
votes
2answers
61 views

For $R$-modules M, is $M\cong R^{\oplus n}\otimes_RM\cong M^{\oplus n}$?

I wanted to explicitly give a bilinear map $R^{\oplus n}\times M\longrightarrow M^{\oplus n}$ when trying to prove that $R^{\oplus n}\otimes_RM\cong M^{\oplus n}$ and ended up with ...
1
vote
0answers
21 views

Make a vector space into a module on ring of polynomials

I'm teaching myself about modules and I came across this exercise that I'm having trouble with (found here: http://www.math.uiuc.edu/~r-ash/Algebra/Chapter4.pdf). The question asks: Let $T$ be a ...
1
vote
3answers
54 views

A conceptual question in ring theory?

What is the main(conceptual) difference between an ideal of a ring and a submodule over a ring?
0
votes
2answers
46 views

For all $R$-modules $N$, $R\otimes_R N\cong N$

Why is this so? This statement is from Aluffi's book Algebra: Chapter 0 and seemingly so trivial that it deserves no proof. So working with the universal property of tensor products, how exactly does ...
1
vote
1answer
15 views

Relationship between idempotents in semisimple ring.

Let R be a semisimple ring with identity. If $e$ and $e'$ are idempotents in R such that $Re \simeq Re'$ then there exists $a \in R^\times$ such that $e' = aea^{-1}$ Attempt I know from a previous ...
-1
votes
1answer
45 views

How can I prove that $\mathbb{C}[x]$ is finitely generated as an $S$-module, where $S$ is a subring of $\mathbb{C}[x]$? [closed]

Suppose $S$ is a subring of the polynomial ring $\mathbb{C}[x]$ where $\mathbb{C}\subsetneq S$. How can I prove that $\mathbb{C}[x]$ is finitely generated as an $S$-module?
0
votes
0answers
10 views

The reduced norm map $\operatorname{Nrd}: K_1(A)\to K^\times$

Let $K_1(A)$ be the Grothendieck $K_1$-group of the category of finitely generated projective $A$-modules where $A$ is a central simple $K$-algebra. I'd be grateful if someone could tell me if the ...
0
votes
1answer
17 views

What is the reduced norm map?

This is a basic question about the reduced norm homomorphism. Let $A$ be a central simple $K$-algebra and $P$ a f.g. projective $A$-module. I know that $\operatorname{End}_A(P)$ is also a central ...
1
vote
1answer
43 views

A free module over $\Bbb C[x]$

Define the square matrix $A$ over $\Bbb C[x]$ by $$ A:= \begin{pmatrix} x & -2x \\ -x^{2} & x^{3}-x \end{pmatrix}. $$ Prove that the submodule $AV=\{ Av \mid v \in V \}$ of the $\Bbb ...
0
votes
1answer
25 views

Why $S\cong A/I$?

Let $A$ a $\mathbb K-$algebra of finite dimension where $\mathbb K$ is an algebraically closed field and let $S$ a simple $A-$module. In the proof of the Schur lemma, it says that since $S\cong A/I$ ...
2
votes
2answers
68 views

$R$ commutative ring with unity , does polynomials with unit leading coefficients of degre s from $0$ to $n$ generate all polynomials of deg $\le n$?

Let $R$ be a commutative ring with unity , consider the polynomial ring $R[x]$ , let $\mathcal P_n:=\{f \in R[x] : f=0$ or $\deg f \le n\}$ , so $\mathcal P_n$ is a finitely generated module over $R$ ...
2
votes
1answer
32 views

Minimum cardinality module for a fixed finite ring

Let $F$ be a finite field and $k$ be a positive integer. Let $M_k(F)$ denote the ring of $k\times k$ matrices. $M_k(F)$ is an $M_k(F)$-module with matrix multiplication, and $F^k$ is an ...
1
vote
0answers
27 views

Does $\phi: A \otimes \mathbb{Q} \to B \otimes \mathbb{Q}$ surj. imply that for $b \in B$, $b = n \phi(a)$ for some $n \in \mathbb{Z}$, $a \in A$?

Let $A$ and $B$ be abelian groups. Suppose that we have a morphism $\phi: A \to B$ and that $\phi \otimes_\mathbb{Z} \mathbb{Q}: A \otimes_\mathbb{Z} \mathbb{Q} \to B \otimes_\mathbb{Z} \mathbb{Q}$ is ...
1
vote
3answers
92 views

$\mathbb{Z}[i]\otimes_{\mathbb{Z}}\mathbb{R}$ is isomorphic to the complex numbers

I am new to tensor poducts (of modules over a commutative ring with identity) and need to understand the following example to continue with the actual exercises in my material. Namely, I need to ...
1
vote
1answer
51 views

Projective module with non-zero annihilator [closed]

Let $M$ be a projective module. Suppose $\operatorname{Ann}_{R} \left(M \right) \neq 0$, where $\operatorname{Ann}_{R} \left( M \right) =\{r\in R : mr = 0, \ \forall m \in M \}$. Then there exists an ...
0
votes
0answers
10 views

Equivalent definitions of primary ideals [duplicate]

In Atiyah-Macdonald, an ideal $I\in R$ is primary if $fg\in I$ implies that $f\in I$ or $g^n \in I$ for some $n$. The ideal $I$ is $P$-primary for a prime ideal $P$ if it is primary and $\sqrt{I}=P$. ...
3
votes
1answer
83 views

Isomorphism between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
1
vote
1answer
27 views

$M_{1} \oplus M_{2}$ is a cyclic $A$-module $\iff \rm{Ann}(M_1)+\rm{Ann}(M_2)=A$ [duplicate]

Let $A$ be a commutative ring with an identity element $1$. An element $x$ in an $A$-module $M$ is called cyclic if $Ax=M$. An $A$-module which has a cyclic element is called cyclic $A$-module. Let ...
0
votes
0answers
15 views

Endowing module with algebra

Given a module $M$ over a ring $R$, is it possible to endow $M$ with an operation $M^{2} \to M$ that turns $M$ to an algebra that is not the trivial $m n \equiv \mathbf{0}$ identically? So far I have ...
0
votes
1answer
57 views

Same kernels for homomorphisms of free modules

Let $f: R^n \rightarrow R^m$ be an isomorphism of free $R$-modules ($R$ commutative with unity) and $\pi_1: R^n \rightarrow R^n/\mathfrak m^n$, $\pi_2: R^m \rightarrow R^m/\mathfrak m^m$ the canonical ...
1
vote
0answers
31 views

Let a,b have the same divisor (content) in an integral domain A. When can I deduce $a/b\in A^\times$?

Given a Noetherian integral domain A and a finitely generated torsion A-module M, we can define the divisor, or content, of M to be $div(M)= \sum_{P, ht(P)=1} \ell(M_P) [P]$, where the sum ranges ...
-2
votes
1answer
20 views

Module is isomorphic to a direct sum of modules, then the length(M)=sum length(M_i)

Is there any proof (or even a counterexample) for: If $M\tilde{=} M_1\oplus ... \oplus M_n$, then it follows for the finite length $l(M) = l(M_1)+...+l(M_n)$. (Modules of a commutative ring with ...
1
vote
1answer
29 views

If every cyclic $R$-module is projective, then $R$ is semisimple.

If every cyclic $R$-module is projective, then $R$ is semisimple. I know how to prove this if the definition of cyclic module is $M=Rm$ where $m$ is an element of $M$. The problem is, we defined ...
0
votes
1answer
22 views

The existence of a non-split composition series in a indecomposable module

Assume that $R$ is a ring with unit and $M$ is a indecomposable left $R$-module with finite length. That is, $M$ has a composition series. Is it true that there is a composition series ...
2
votes
1answer
21 views

Torsion in module over the ring of convergent power series

I need to understand a passage from a paper which I don't quite understand. Let $M$ be a module over the ring $\mathbb C\{t\}$ of convergent power series. We want to show that $M$ is torsion-free, ...
0
votes
0answers
9 views

for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\in R$.

Let be a $R$ ring with a identity. An $R$-module $A$ is injective iff for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every ...
0
votes
0answers
13 views

Endomorphism of Central Simple Algebra

Let $A\in\mathscr{C}(F)$, i.e. $A$ is a central simple algebra. Show that $\text{End}_F(A)\cong M_n(F)$ as $F$-algebras where $n=\dim A$. My idea is to consider $A$ as a $F$-vector space, then ...
0
votes
1answer
24 views

How can I prove that $2\mathbb{Z}$ is NOT a direct summand of $\mathbb{Z}$, as $\mathbb{Z}$-modules?

I know that $\mathbb{Z}=2\mathbb{Z}+\{0\}$ (with "+" I mean "direct sum"). Is this enough to prove that $2\mathbb{Z}$ is NOT a direct summand of $\mathbb{Z}$, as $\mathbb{Z}$-modules?
0
votes
1answer
19 views

Confused about order in Opposite Algebra

I am facing confusion in the "order of multiplication" regarding the opposite algebra $B^o$ in the following working: Define a right $B^o$-module $M_\phi$ where $M_\phi=M$ via $m\cdot b=m\phi(b)$. ...
2
votes
2answers
30 views

Elementary tensors

Let $G,H$ be $R$-modules, and $G \otimes H$ be it's tensor product. I can't prove it and I suspect it's false that any element $\tau \in G \otimes H$ can be written as $\tau = g \otimes h$ for some ...
2
votes
1answer
31 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
0
votes
2answers
38 views

$M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ for cyclic modules $M,N$?

Is it true that $M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ (considering $\mathbb{Z}$ as a $\mathbb{Z}$-module) if $M,N$ are cyclic $\mathbb{Z}$-modules (i. e. generated by one element)? I would ...
1
vote
2answers
50 views

if $a_1 \cdot b \equiv a_2 \cdot b (\text{mod }n)$ then $a_1 = a_2$

Assume $a_1 \cdot b \equiv a_2 \cdot b (\text{mod }n)$ and also $a_i^{\frac{n-1}{2}}$ mod $n \in \{1, n-1\}$ and $b^{\frac{n-1}{2}}$ mod $n \notin \{1, n-1\}$, I saw the following which proves $a_1 = ...
1
vote
0answers
35 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
-1
votes
2answers
37 views

Showing $R\otimes M \cong M$ for $R$-modules $R,M$

How to see that $R \otimes M \cong M$ if $R$ and $M$ are $R$-modules (with $R$ being a commutative ring with unity)? I thought about defining $f: R\otimes M \rightarrow M$ by $(r,m) \mapsto rm$. ...
1
vote
2answers
18 views

Providing non trivial module morphisms

I'm starting to study modules, and I would like to get some counterexamples to naive ideas one has in the first approach to the subject. Does there exist an ideal I in A and a morphism $f:I ...
4
votes
0answers
88 views

Testing if a submodule is free

This is hopefully a very simple question. In Gröbner Bases in Commutative Algebra by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, $S=K[x_1\ldots ...
1
vote
0answers
15 views

Rank of a locally free $\mathbb Z[G]$- module

Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$. Here $\mathbb Z_p[G]=\mathbb Z_p\otimes_\mathbb ...
1
vote
0answers
15 views

How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...
1
vote
1answer
37 views

Tensor product (confusing question)

Let's say I have a tensor product $A\otimes B$ of algebras $A,B$. I have a linearly independent subset $S\subseteq A\otimes B$ such that $span(S)\cong C$, where $C$ is another algebra. Similarly, ...
0
votes
0answers
24 views

Composition series of a regular module.

Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$ e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0. $$ I am trying to find the composition series for ...
1
vote
2answers
66 views

Corollary to Lemma of Nakayama

In Matsumura's Commutative Algebra there is the following Corollary to the Lemma of Nakayama: Let $A$ be a ring, $M$ an $A$-module, $N$ and $N'$ submodules of $M$, and $I$ an ideal of $A$. Suppose ...
2
votes
1answer
62 views

A better way of showing the set of all $m\times n$ matrix is an $R$ module.

Let $R$ be a ring and $m$ and $n$ be any positive integers. For any $a\in R$ and $A=(a_{ij})\in M_{m\times n}(R)$ define $aA=(aa_{ij})$. Then prove that $M_{m\times n}(R)$ is an $R$-module. Edited: ...
2
votes
1answer
36 views

Every module free implies no nonzero maximal ideal.

Show that given a ring $R$ with identity such that every $R$-module is free, then $R$ has no nonzero maximal ideals. I only know that every ideal of $R$ is a direct summand of $R$. Is it ...
0
votes
1answer
28 views

Show that $\operatorname{End}_A(P)$ is a central, simple $K$-algebra if $P$ is a f.g. projective $A$-module

Let $A$ be a central, simple $K$-algebra and let $P$ be a finitely generated projective $A$-module. I want to show that the endomorphism ring $\operatorname{End}_A(P)$ is also a central, simple ...
0
votes
3answers
39 views

why if $a \text{ mod } p = -1$ then $a \text{ mod } p = p-1$

This seems very simple and obvious but I can't prove it. why if $a \text{ mod } p = -1$ then $a \text{ mod } p = p-1$? thank you.
2
votes
2answers
27 views

Finite dimensional representations of the Weyl algebra in characteristic $p>0$

I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra ...
1
vote
3answers
33 views

Annihilator of modules [duplicate]

If $A$ is an $R$-module, I am having difficulty proving that $A$ is also a well-defined $R/ann(A)$-module with $(r+ann(A))a=ra$.
1
vote
0answers
18 views

A illuminating theorem on general modules over a ring

I'm going to do a one hour presentation on modules, free modules and projective modules over a ring. While most of my motivation for studying them comes from my interest in starting homological ...
2
votes
0answers
17 views

Injective homomorphism from an ideal to a torsion free module over a Dedekind ring

Let $M$ be a nonzero torsion free module over a Dedekind ring $R$ and $I ⊂ R$ an ideal. Show that there is an injective $R$-module homomorphism $I → M$. My solution is to take a nonzero element ...