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

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2answers
33 views

Showing that an epimorphism to a free module of finite rank splits

Let $M$ be an $R$-module and let $F$ be a free $R$-module of finite rank. Let $\phi : M \to F$ be an epimorphism. Then show that $M$ has a submodule $F' \cong F $ such that $M=F' \oplus \ker\phi$. ...
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0answers
63 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace, I would ...
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1answer
35 views

A Problem about Spliting of Exact Sequence

Module Case Let $$0\rightarrow N \stackrel f \rightarrow G\stackrel g \rightarrow Q\rightarrow 0$$ be the exact $R$-module sequence iff (1) there exists $u:G\rightarrow N$ which satisfies $u\circ ...
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0answers
50 views

'Finitely generated modules' versus 'Finitely generated algebra'

I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page ...
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1answer
22 views

Calculating the dual module

Is the dual $\mathbb{Z}$-module of $\mathbb{Z}/n\mathbb{Z}$, that is ${\rm Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\mathbb{Z})$ isomorphic to $\mathbb{Z}/n\mathbb{Z}$. Looking at it briefly I think ...
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1answer
77 views

Relation between Jacobson radical and composition series

Let $R$ be a not necessarily commutative ring with 1. Suppose $R$, viewed as a right $R$-module, has a finite composition series with non-isomorphic composition factors. Prove that the Jacobson ...
2
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2answers
132 views

Representation theory of infinite groups?

I am familiar with the representation theory of finite groups (at least of the symmetric groups over the field of complexes) And I know that the group algebra of an infinite group is not semisimpe ...
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1answer
32 views

Incomplete Proof from Neukirch about modules and algebraic integers

There is a theorem in Neukirch which says If $L | K$ is separable and $A$ is a principle ideal domain, then every finitely generated $B$-submodule $M \neq 0$ of $L$ is a free $A$-module of rank ...
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2answers
42 views

If $N$ is an $R/I$-submodule of $M$ can we view $N$ as an $R$-submodule of $M$?

Let $R$ be an integral domain, $M$ an $R$-module and $I\subseteq \mathrm{Ann}(M)$ an ideal of $R$. $N$ is an $R/I$-submodule of $M$ (as $R/I$-module). Can we view $N$ as an $R$-submodule of $M$ ...
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2answers
94 views

Module of $R$-valued functions on an infinite set is not countably generated

Let $R$ be an integral domain and $X$ be an infinite set. Let $R^X$ be the set of all functions $f: X \rightarrow R$, viewed as an $R$-module in the usual manner: for $\alpha \in R$, $\alpha f: x \in ...
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2answers
95 views

If $k\subset R\subset k[x]$, then $R$ is Noetherian?

Is there a way to prove that any subring $R$ of the polynomial over a field $k$ such that $k\subset R$ is Notherian without appealing to integral extensions, Eakin-Nagata, etc.? The reason I ask is ...
2
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1answer
37 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
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1answer
73 views

Prove $A_\mathfrak{p} \otimes_A B_\mathfrak{q} = B_\mathfrak{q}$, where $\mathfrak{q}$ prime in $B$

$\require{AMScd}$ Hi, I think I have the answer for this question, but I'm not sure if it's correct. So I would be very glad if someone could have a quick look through it. Let $A$, $B$ be ...
5
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1answer
44 views

holonomic D-modules

I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables, ...
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0answers
105 views

Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence \begin{equation} 0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0. \end{equation} ...
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1answer
43 views

Computing the kernel of a homomorphism from a free $\mathbb{Z}$-module

Given a finitely generated free $\mathbb{Z}$-module $N$ and a homomorphism $\varphi:N\to M$, is there an easy way to compute the kernel of $\varphi$? Since the kernel is also a free ...
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0answers
38 views

Dual of polynomial ring

Consider the free k-algebra $k[x_i]_{i \in I}$ indexed by I. Then is: $Hom_{k-Mod}(k[x_i]_{i \in I},k) \cong k[x_i]_{i \in I}$
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0answers
41 views

Enveloping Algebra equal to algebra

Let $R$ be a unital associative ring, $A$ be an associative $R$-algebra of finite dimension, and $A^e$ its enveloping algebra. What are the requirements on $A$, so that $A^e \cong A$ (as ...
2
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1answer
50 views

Duals of a finite dimensional eveloping coalgebra

Let $C^e$ be the enveloping $k$-coalgebra of a $k$-coalgebra $C$ and denote by ${C^e}^{\star}:=\mathrm{Hom}\,_{k}(C^e,k)$. Then is ${C^e}^{\star} \cong {C^{\star}}^e$?
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1answer
36 views

Epimorphism that is not monomorphism from $M\rightarrow M$

I have just finished an exercise where I prove that if $M$ is a module with acc then any epimorphism $f:M\rightarrow M$ but be an isomorphism. I then had a think about examples of non-noetherian ...
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0answers
37 views

Right approximation of a module [duplicate]

This question has confused me. I appreciate anybody who can help or give a reference. Let $R$ be a ring and $A$, $X$, and $M$ be $R$-modules. We say that a map $f:M\to A$ is an $X$-minimal right ...
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1answer
44 views

How do we compute $\mathbb{Z}^2/(n, m)$?

I have been trying to compute quotients of this form (as modules). Does anyone have a quick method of doing this? I'm sure I'm missing something obvious. Any help is appreciated!
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3answers
83 views

Tensor products and monomorphisms of $A$-modules.

This problem addresses the same question that has been asked in $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$, which is exercise 2.11 of Atiyah and Macdonald's Introduction to ...
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0answers
32 views

Dimension with respect to a module

This question has confused me. I appreciate anybody who can help or give a reference. Let $R$ be a ring and $A$, $X$, and $M$ be $R$-modules. We say that a map $f:M\longrightarrow A$ is an ...
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0answers
17 views

Simultaneous decomposition of modules over Dedekind domains

In the paper "Almost diagonal matrices over Dedekind Domains" by L. Levy a specific decomposition of modules over Dedekind rings $D$ is used: Let $M$ be submodule of $D^n$, then there exists a ...
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2answers
44 views

Is vectorspace trivial under these conditions?

Let $R$ be a ring. Looking for a left $R$-module free over abelian group $A$, I arrived at $\left|R\right|\otimes A$ with $r.\left(s\otimes a\right)=rs\otimes a$ where $\left|R\right|$ denotes the ...
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1answer
74 views

Representation theory approach VS Module theory approach?

Given an associative algebra $A$, there is a correspondence between representations of $A$ and left $A-$ modules. Thus, one can study the representation theory of an associative algebra via its left ...
3
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2answers
43 views

Is it possible that a scalar times a submodule is not a submodule?

Does there exist a ring $A$ and an $A$-module $X,$ such that for some $a \in A$ and some submodule $Y \subseteq X$, it holds that $aY$ is not a submodule? If $A$ is commutative, this is clearly ...
2
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1answer
133 views

Polynomial is zero for induced mapping of rings

Let $R$ be a commutative ring, and $M$ a finitely-generated free $R$-module. Let $\phi:M\rightarrow M$ be an $R$-linear map, and $P_\phi(X)$ the characteristic polynomial of $\phi$. Let ...
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0answers
94 views

$\mathrm{Hom}$ and $^*\mathrm{Hom}$ for graded modules: Exercise 1.5.19(f) of Bruns-Herzog

Assume $R$ is a graded ring and $M$ and $N$ graded modules. Denote by $^*\mathrm{Hom}_R(M,N)$ the set of all homogeneous $R$-linear maps from $M$ to $N$. How can I prove that if $M$ is finitely ...
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3answers
54 views

Showing that simple modules over finitely generated commutative algebras are 1-dimensional and isomorphic to…

Let $A= \Bbb{C}[x_1, ... ,x_n]$. a) Show that every simple $A$-module is 1-dimensional and is isomorphic to $\Bbb{C}[x_1, ... , x_n]/(x_1 - a_1, ... , x_n - a_n)$ for some $a_1, ... , a_n \in ...
2
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1answer
48 views

Matrix algebras

Let $k$ be any field, then we know that every finite dimensional semi simple algebra $A$ is isomorphic to a direct product of matrix algebras with entries over a division ring. Assume that we require ...
1
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1answer
62 views

What is a regular FG-module?

What is meant by a regular $FG$-module. $G$ is a group and I believe $F$ is supposed to be a field. I'm completely confused by this concept on a question sheet and I can find lots of uses of the ...
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2answers
95 views

If for $A$ a commutative nonzero ring $A^m ≅ A^n$ as $A$-modules, then $m = n$

This is the problem I need to solve: Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$. The book I got this problem from suggests using the following method to solve it: Let ...
5
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1answer
36 views

Is it true that $\operatorname{colim}\operatorname{Hom}_{R_i}(M,N)$ is isomorphic to $\operatorname{Hom}_{\lim R_i}(M,N)$?

Suppose we have a system of ring homomorphisms $\cdots \to R_{i+1} \stackrel{f_i}{\to} R_i \to \cdots \stackrel{f_0}{\to} R_0$ (we may assume all the maps are injective, but that's not necessary for ...
2
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0answers
18 views

Smooth algebras and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
2
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2answers
85 views

Show that $Ker(f)$ is finitely generated, when $f: M \rightarrow A^n$ is a surjective A-module homomorphism and M is finitely generated

This is the problem I am attempting: Let $M$ be a finitely generated A-module and $f: M \rightarrow A^n$ a surjective homomorphism. Show that $Ker(f)$ is finitely generated. Following a hint in ...
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0answers
41 views

Modules,rings and definitions

Is there a source available with (almost) all definitions from ring and module theory,all in ONE place without theorems.There are books on module theory where are freely used unusual notions like ...
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1answer
65 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
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1answer
84 views

Free Module over the Integers

Without using the fact that submodules of free modules over PIDs, how can one go about showing that the real numbers don't form a free Z module
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0answers
30 views

Quotient of projective indecomposable modules by their radicals

Assume $\Lambda$ is an artin algebra and let $P$ be an indecomposable finitely generated projective $\Lambda$-module. Let $J(P)$ denote the Jacobson radical of $P$ (which, in this setting, is the ...
2
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2answers
43 views

If $N\cap rM=rN$ for all $r\in R$, then is $M=N\oplus K$ for some $K$?

Suppose $M$ is a finitely generated free module over a principal ideal domain $R$, and $N$ a submodule. Why does the condition $N\cap rM=rN$ for all $r\in R$ implies that $M=N\oplus K$ for some ...
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1answer
52 views

Is every submodule of a free module of finite rank over a PID a direct summand?

Suppose $F$ is a free module of rank $n$ over a PID, with $N$ a submodule. Is $N$ always a direct summand of $F$? I think the answer is yes, $N$ is also free of rank $m\leq n$ since we are working ...
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0answers
45 views

Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
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1answer
36 views

Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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1answer
23 views

Rank of a subgroup of $\mathbb{Z}^3$ given a generating set $(2,-2,0)$, $(0,4,-4)$, and $(5,0,-5)$?

Is there some standard approach to finding the rank of a subgroup given a generating set? In particular, I'm considering the subgroup of $\mathbb{Z}^3$ generated by $v_1=(2,-2,0)$, $v_2=(0,4,-4)$, ...
6
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0answers
78 views

Two ideals both alike in dignity, in fair Paris where we lay our scene.

Let $A$ be an integral domain. I have to show that two ideals $\mathfrak a$ and $\mathfrak b$ are isomorphic as $A$-modules if and only if there exist $a$ and $b$ such that $a\mathfrak b=b\mathfrak ...
-1
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1answer
35 views

Free Module over a tensor product

Suppose that $A$ and $B$ are $\mathbb{K}$-algebras for some field $\mathbb{K}$. Is $B$ free over $A \otimes_{\mathbb{K}} B$ ?
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0answers
137 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...
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3answers
134 views

Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...