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
39 views

Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
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1answer
30 views

Basis of a subset of finitely generated torsion free module

Based on the comments of rschwieb's answer in this question asked recently: Can we contruct a basis in a finitely generated module. If $M=\langle e_1,\ldots,e_n\rangle$ is a finitely generated ...
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2answers
24 views

Isomorphism between modules

Let $R$ be a PID , r $\in R, r\neq 0, n\in \mathbb{N}$ Prove that there is an isomorphism $R/rR \simeq Rr^n/Rr^{n+1}$
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1answer
39 views

Extension and restriction of scalars and their relation to the identity functor

Let $R$ and $S$ be rings with $R \subseteq S$, and let $_S M$ be a left $S$-module. If we restrict our scalars to $R$, we naturally have a left $R$-module structure on $M$, $_R M$, given by ...
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0answers
20 views

Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
1
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1answer
27 views

Exact Sequences of R-Modules

Here's a lemma in A Course in Ring Theory by Passman. In the proof it is mentioned, "But the kernel of the combined epimorphism $P\rightarrow B\rightarrow C$ is clearly equal to $E$". I don't ...
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1answer
15 views

Projective Dimension and Supremum

Here is a lemma that appears in A Course in Ring Theory by Passman. In the last section of the proof the writer shows that, $\mbox{pd }A_i\leq n\iff \mbox{pd }A\leq n$ and finishes the proof. I don't ...
2
votes
1answer
82 views

Is always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?

Let $A$ be a commutative ring. Let $f \in A$. Let $A_f= A\left [ \frac{1}{f}\right ]$. Let $\hat{A}$ the $f$-adic completion of $A$. Is it always (even when $A$ is not noetherian) true that $$\hat{A} ...
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0answers
49 views

Localization at prime ideal $(x_1, x_2, \dots)$ of infinitely many variables

I'm trying to solve a homework exercise which starts off: Let $k$ be a field, and let $A = k[x_1, x_2, \dots]$ be a polynomial ring in infinitely many variables. Let $\mathfrak p \subseteq A$ be ...
0
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1answer
24 views

Quotient Module is Not Free

Let R be a commutative ring with a multiplicative identity. $a\in R, a \not = 0$, prove that $R/(a)$ is not a free R-module. My initial attempt was to say that R/(a) is not torsion free, thus it ...
2
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1answer
45 views

Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
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0answers
52 views

Is every module a direct limit of cyclic modules?

I want to show that $M$ is $A$-flat is equivalent to $Tor_1^A(M,A/I)=0$ for every finitely generated ideal $I$. I want to show $Tor^A_1(M,N)=0$ for any $A$-module $N$. Is every module a direct ...
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0answers
14 views

Group ring is necessarily free as a module

I have defined the group ring $R[G]$ to be the set of all liner combinations of the form $$\sum_{g\in G}r_g \,g$$ where $g\in G$ ($G$ any group) and $r_g\in R$ ($R$ any ring) with $r_g\neq0$ only for ...
1
vote
1answer
15 views

Direct sums, submodules and simple tops

Let $A$ be a finite dimensional algebra over some field. Suppose $X_1$ and $X_2$ are $A$-modules, both with simple top (that is, $X_i/ (\operatorname{Rad} X_i)$ is simple for $i=1,2$). Let $K$ be a ...
3
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2answers
63 views

Question concerning $M_1\cap K=M_2\cap K$ and $M_1+K=M_2+K$

I have a ring $R$ with $K\le M$ and submodules $M_1,M_2$. If we have that: $$M_1\cap K=M_2\cap K \text{ and } M_1+K=M_2+K$$ can we conclude that $M_1=M_2$? I don't think that this is true ...
3
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1answer
36 views

Existence of Boundary Homomorphisms for Cohomology

I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of $P_0 = ...
2
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0answers
50 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
2
votes
1answer
17 views

Modules over a product of Algebras

Suppose $A = A_1 \times A_2$ where $A_1$ and $A_2$ are associative algebras (Not necessarily unitary). Show that any $A$-Module $T$ is isomorphic to $M \oplus N$ where $M$ is an $A_1$-module and $N$ ...
3
votes
0answers
30 views

Complements of semisimple rings

Let $R$ be a semisimple ring, let $M_{R}$ be a right-module then every right submodule of $M$ has a complement in $M$. Now assume $M$ is an $R$-bimodule with $R$ semisimple. Then every right ...
2
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0answers
103 views

Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
3
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2answers
47 views

$\mathbb{Z}/n\mathbb{Z}$ is not flat

On the flat module Wikipedia page, it's stated that $\mathbb{Z}/n\mathbb{Z}$ is not flat over $\mathbb{Z}$. But I don't understand their explanation of why. It is said that ...
0
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1answer
52 views

Doubt about the tensor product

Suppose $M$ is an abelian group and $F(X)$ is the free abelian group over $X$. Is it true that any element of $M\otimes F(X)$ can be written as a finite sum $$m_{1}\otimes x_{1}+ \cdots+m_{n}\otimes ...
0
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1answer
26 views

Is quotient module finitely generated?

Suppose $R$ be any ring containing left ideal $I$. Then $I$ is submodule of $R$, so $R/I$ is R-module. My question is, is $R/I$ always a finitely generated?
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1answer
43 views

Simple $R$-module

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true: 1) $N$ has a finite number of submodules. 2) $\operatorname{Hom}_R(N,N)$ is a division ring. 3) ...
0
votes
1answer
45 views

associated $\mathbb{C}[t]$- module is cyclic iff cyclic vector exists

I'm stuck on a part of a question: if $T : V \rightarrow V$ is a linear endomorphism of a $\mathbb{C}$-vector space $V$, then the associated $\mathbb{C}[t]$- module is cyclic (that is $V ...
2
votes
1answer
60 views

A module over an algebra. Is it a vector space?

Let $A$ be an algebra over a field $k$. I would like to know if my understanding of the following correct or not. What I want to clarify is the definition of a module $M$ over $A$. I know the ...
0
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0answers
35 views

characterization of simple module [duplicate]

Let $(R,M)$ be a local ring and $Rx$ a nonzero summand of $M$. Then $Rx$ is a simple module iff $x^2=0$, where $M$ is a maximal ideal and $Rx$ be the cyclic submodule of $R$.
2
votes
1answer
29 views

Tensor product of certain algebras

If S is an $R$-algebra of finite dimension, and $A$ is an algebra of infinite dimension, then is $S \otimes S \otimes A \cong S \otimes A$?
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0answers
46 views

Problem about tensor products and modules over a group algebra

Can anyone help me with hints for these problems? I would appreciate it a lot. 1) Supose $G$ is a group and $k$ is a field. If $U,V$ and $W$ are $kG$-modules then $\operatorname{Hom}_{kG}(U\otimes ...
8
votes
2answers
64 views

Zorn's Lemma in noetherian modules

For noetherian modules, we have in particular the equivalent definitions that the Ascending Chain Condition holds and that every nonempty subset of submodules has a maximal element. Now I can prove ...
1
vote
1answer
33 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
0
votes
1answer
29 views

Showing that finitely generated module forms a torsion quotient.

I have that $R$ is a domain and $M$ is a finitely-generated $R$-module. I am trying to show that there is a free submodule $F$ such that $M/F$ is a torsion $R$-module. I have the fact: If $X$ is a ...
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0answers
37 views

Module in $\mathbb{Z}$ and characteristics

I'd like to show some basic characteristics for a module in $\mathbb{Z}$. The module in Z has been defined as: "A subset $\not0 \not = M \subseteq \mathbb{Z}$ is called module, if M is closed under ...
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0answers
18 views

$R$ ring ,$ I,J$ sets and $I$ infinite. if $R^{(I)}$ izo of modules with $R^{(J)}$ then $| I | = | J |$.

I have to prove that. I know we have 2 cases. 1) If $| I | > | J |$ 1.1 $J$ infinite 1.2 $J$ finite 2) if $| I | < | J |$ this implies case 1.1 Please help me at case 1.2
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0answers
21 views

Extension of Scalars as Equivalence Classes

Given a ring morphism $f\colon R\to S$, there is a functor $f_!\colon RMod\to SMod$, $N\mapsto S\otimes_R N$ called "the extension of scalars". My confusion arises from a statement in nLab's article ...
2
votes
1answer
32 views

Localization of a module is zero implies the multiplicatively closed subset contains a single element annihilating the module

I need to show that if $S$ is a multiplicatively closed subset of a ring $A$, $M$ is a finitely generated $A$-module, and $S^{-1}M = 0$, then there exists a single element $s$ in $S$ so that $sM = ...
0
votes
1answer
23 views

modules finite congenerated are closed under extensions

I have to prove some properties about modules finite cogenerated, I´ve already prove that mmodules finite cogenerated are closed under submodules, finite direct sums, but I can´t see how to prove that ...
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1answer
32 views

The sum of all right ideals isomorphic as modules to a simple module is an ideal

I could use some help on the following problem. Let R be a ring. (a) If $r \in R$ and $U$ is a minimal right ideal of $R$, show that either $rU=0$, or that $rU$ and $U$ are isomorphic right ...
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1answer
73 views

$m$-primary ideal and $M\otimes_{A} A/m \neq 0$

Let $A$ be a commutative local ring with maximal ideal $m$. Let $M$ be a (not necessarily finitely generated) $A$-module. Let $x_{1},\dots,x_{n}$ be an $M$-regular sequence such that ...
4
votes
1answer
47 views

Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion.

Given an integral domain $R$, and a left torsion $R$-module $A$ (i.e. $\forall{a}\in A,\exists{r}\in R$ such that $ra=0$) how would you show that $\mathrm{Tor}_n^R(A,B)$ is also a torsion $R$-module?
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0answers
90 views

When is $\operatorname{gr}_I (M)$ finite?

When is $\operatorname{gr}_I (R)$ (I mean associated graded ring of $I$) finite? When is $\operatorname{gr}_I (M)$ finite? ($M$ is an $R$-module.)
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vote
1answer
48 views

How to calculate the tensor product?

This question might be stupid for you.I ask because I have no clue about it. I don't really understand what is tensor product,although I know its definition. I have search what is tensor product,so I ...
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0answers
35 views

Is the functor category of algebras into modules locally small?

Let $R$ be a ring. Is $[{}_R Alg, {}_R Mod]$ locally small?
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0answers
49 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
5
votes
1answer
52 views

Jordan-Holder theorem for modules?

Let $A$ be a finite dimensional algebra over some field $ k$. I think from Jordan-Holder Theorem, one might be able to claim that every simple $A-$module occurs in the series (by this I mean it is ...
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0answers
19 views

Linear Differential Equation with Quasiperiodic Coefficient: A Question

I am reading the book Synergetics. The following passage is from the book: Assume that in the differential equation $\dot{q}= a(t)q$, where $a(t)$ is quasiperiodic, i. e., we assume that $a(t)$ ...
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1answer
76 views

“M is reflexive” implies “M is MCM”. Is the converse true?

Let $(R,m)$ be a local integrally closed domain of dimension $2$. Let $M$ be a nonzero finitely generated $R$-module. We know that "$M$ is reflexive" implies "$M$ is MCM". Is the converse true? (By ...
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1answer
33 views

Why these two propositions have different requirements

Proposition 2.18 is similar to 2.19. Why we need $N$ flat in 2.19? What's the difference between 2.18 and 2.19?
1
vote
1answer
51 views

Intuition for the fact that, in a vector space V over a field F, av = 0 $\implies$ a = 0 or v = 0. (a $\in$ F, v $\in$ V).

I have no trouble proving this: Let av = 0. If a = 0 then then we are done. Otherwise, there exists $a^{-1} \in F$ such that $a{^-1} a = 1$. Multiplying both sides of the equation by $a^{-1}$ gives ...
0
votes
1answer
68 views

The ideal $I=(3,2+\sqrt {-5})$ is a projective module

Let $R=\mathbb Z[\sqrt{-5}]$ and $I=(3,2+\sqrt {-5})$ be the ideal generated by $3$ and $2+\sqrt{-5}$. I'm trying to prove that $I$ is a projective $R$-module. I'm using the lifting property ...