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

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84 views

Direct-Sum Decomposition of an Artinian module

Let $R$ be a commutative Noetherian ring. Suppose $M$ is a finitely-generated non-zero Artinian $R$-module. Question: How can we prove that there are maximal ideals $m_1 , m_2 , \ldots , m_n$ such ...
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1answer
72 views

Example: $ u\in A \otimes_R B$, but $u \neq a \otimes b $ for any $ a \in A, b \in B.$ [closed]

Give an example to show the following may actually occur for suitable ring $R$ and modules $A,B$: $ u\in A \otimes_R B$, but $u \neq a \otimes b $ for any $ a \in A, b \in B.$
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2answers
29 views

Annihilator of extension of scalars vs. the extension the annihilatar

Let $A,B$ be commutative rings with 1, $f:A\to B$ a morphism of rings, $M$ an $A$-module, and $M_B=B\otimes_AM$ the extension of scalars. Then is it the case that $\text{Ann}(M)^e=\text{Ann}(M_B)$? ...
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1answer
75 views

Defining a sheaf of differential operators

I have two questions related to the coordinate-free definition of $\mathcal{D}$-modules provided in Ginzburg's notes (see pages 24-25): Exercise 2.1.13 says that Diff($M,M$) is almost-commutative, ...
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1answer
76 views

$R$ noetherian, $I$ injective $R$-module $\Rightarrow$ $S^{-1}I$ is injective over $S^{-1}R$

I am trying to prove that if $R$ is a noetherian ring, $S$ a multiplicative part and $I$ an injective $R$-module, then $S^{-1}I$ is an injective $S^{-1}R$-module. So far I thought: I reduce to check ...
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0answers
45 views

$M\simeq X\oplus Y$ and $M/X \simeq Y$

I am trying to prove that if $M,X,Y$ are modules over a ring $D$, then $M\simeq X\oplus Y$ $\Longleftrightarrow$ $M/X \simeq Y$ "$\Longrightarrow$'' I consider the map $f_1:x\in X \rightarrow ...
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0answers
92 views

change the matrix when we extend the field

Let $M$ be an $F_pC_q$- module represented by the matrix $$\left( \begin{matrix} a & b\\ c & d \end{matrix}\right)$$ i.e., $m_1 g=am_1 + bm_2$ and $m_2g=cm_1 + dm_2$ where g is the ...
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1answer
28 views

On the Semisimplicity of a Permutation Module given by a Transitive Group Action

Let $G$ be a finite group acting transitively on a finite set $\Omega$. Let $K$ be field such that its characteristic divides $|\Omega|$. Is it true that $K\Omega$ is not semisimple? I think this ...
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1answer
75 views

Extension problem

Let $R$ be the ring consisting of the upper triangular $3\times3$ matrices over $\mathbb{Q}$ and $M_{1},\,M_{2}$ two $R$-modules defined as follows: for $A=(a_{ij})\in R$, $q\in \mathbb{Q}$ ...
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1answer
38 views

definition of product of modules

I have been given this definition of a product between modules: If $I$ is an indexing set with $M_i$ as an $R$-Module then the product $\prod \limits_{i \in I} M_i$ is defined as the set consisting ...
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1answer
23 views

a question about artinian modules

Let M be a modules. Are the following equivalent. i) M is artinian ii) every factor of $M$ is finitely generated. Yours,
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1answer
31 views

Modules isomorphism

Studying vector spaces, we can findthe well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules? Thanks guys!
2
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0answers
53 views

torsion free RG-module

Let $R$ be a PID, $G$ be a cyclic group, $M$ be an $RG$-module and $N$ be a submodule of $M$. How can we test whether $M/N$ is torsion free as an $RG$-module or not? (I know how if we consider it as ...
2
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1answer
48 views

Isomorphism between modules

Let $R$ be a ring with identity and $e\in R$ is an idempotent. It is well-known that $Hom_R(R,M)$ is isomorphic to $M$ as right $R$-modules, for any right $R$-module $M$. Is it true that $Hom(eR,M)$ ...
4
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1answer
75 views

Is $((a \mod n) + (b \mod n) ) = (a + b) \mod n$?

As we know $(a + b) \mod n = ((a\, \bmod\, n) + (b\, \bmod\, n))\, \bmod\, n$ Is their reversal also true like this $((a \mod n) + (b \mod n)) = (a + b) \mod n$. If not then what could be its ...
3
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1answer
83 views

Exactness of Hom Functor

The picture above is from Dummit and Foote, Third Edition, Chapter 10. In the text the authors claim that the sequence given by $ Hom $'s is exact if and only if there is a bijection $ F ...
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0answers
35 views

Equivalence of short exact sequences

The above image is from the book of Dummit and Foote, Edition 3. In the fourth paragraph, the authors claim that "equivalences involving the same extension module $ B $ are automorphisms of $ B ...
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1answer
44 views

Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might ...
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1answer
69 views

solving mod equation

I am attempting to solve $r_1$ in this equation: $$m + xr \equiv m_1 + xr_1 \pmod q$$ This is what I derived at: $$m-m_1 + xr / x \equiv r_1 \pmod q$$ I proceed to sub these with the necessary ...
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1answer
86 views

A non flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0$ for all $n\ge 1$

I want to find a non-flat $R$-module $M$ with $\operatorname{Tor}_{n}^R(k,M)=0 \,\, \forall n\ge 1$, where $R=k[x,y]/(xy)$ and $k$ is field.
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1answer
24 views

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$.

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$. If $M$ is a finite vector space over $F$, then neither do ...
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1answer
59 views

Endomorphisms of a ring

Let $R$ be a ring with identity and let $R^n=P⊕P'$ be a direct sum decomposition with right $R$-modules as its components. We take $e\in\operatorname{End}(R^n_R)$ as the projection of $R^n$ onto $P$, ...
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1answer
27 views

A bimodule homomorphism

$\def\Hom{\operatorname{Hom}}$ Let $R$ be a ring and $P_R$ be a right $R$-module. Set $Q=\Hom_R(P,R)$ and $S=\Hom_R(P,P)$, both operating on the left of $P$. This makes $P$ into an $(S,R)$-bimodule. ...
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1answer
57 views

Integral closure of a PID is torsion free

Can anyone explain me why the integral closure of a PID $A$ in a separable finite extension of its fraction field is a torsion free $A$-module? I know that it is a finitely generated A-module ...
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2answers
40 views

How can $\Bbb{Z}/10$ be viewed as a $\Bbb{Z}$-module?

How can $\Bbb{Z}/10$ be viewed as a $\Bbb{Z}$-module? For example, when I compute $5.\overline{5}$, where $5\in\Bbb{Z}$ and $\overline{5}\in\Bbb{Z}/10$, is this equal to $5$?
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3answers
71 views

Understanding the Definition of the Tensor Product of Chain Complexes

The tensor product of chain complexes (of $R$ modules) $C_\bullet ,D_\bullet$ is defined as $$(C_\bullet \otimes D_\bullet )_n = \bigoplus_{i+j=n} C_i \otimes_R D_{j}$$ I understand this definition ...
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2answers
90 views

Tensor-Hom Adjunction In Monoidal Categories?

Is there a generalization of the tensor-hom adjunction to monoidal categories, or is it a special property of $\mathsf{Mod}$-$R$?
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2answers
74 views

Proof of the properties of tensor product

On page 25 of Atiyah-Macdonald "Introduction to commutative algebra", the author says that "We shall never again need to use the construction of the tensor product given above and the reader may ...
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2answers
64 views

$\Bbb{Z}[X]/\langle f\rangle$ is a finitely generated $\Bbb{Z}$-module

Let $f$ be a monic polynomial in $\Bbb{Z}[X]$. Show that $\Bbb{Z}[X]/〈f〉$ is a finitely generated $\Bbb{Z}$-module. I don't even know how to start. If $g\in\Bbb{Z}[X]/〈f〉$, we are trying to find ...
3
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2answers
49 views

$\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module

I'm trying to show that $\Bbb{Q}$ is not a finitely generated $\Bbb{Z}$-module. Assume to the contrary that $$\Bbb{Q}=\Bbb{Z}\dfrac{a_1}{b_1}+...+\Bbb{Z}\dfrac{a_n}{b_n}$$ where $a_i,b_i\in\Bbb{Z}$. ...
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1answer
40 views

Isomorphism of quotient of direct sum modules

Let $M, N, M'$ and $N'$ be R-modules. If $M'$ and $N'$ are submodules of both $M$ and $N$ then is it true that \begin{equation} \frac{M}{M'} \oplus \frac{N}{N'} \cong \frac{M \oplus N}{M' \oplus N'} ...
3
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1answer
41 views

Homomorphism from a finitely generated module to a direct sum of modules

Let $R$ be a commutative ring with unit. If $M$ and $N_i$ are arbitrary $R$-modules, the module $\operatorname{Hom}_R(M,\bigoplus_{i\in I}N_i)$ is not isomorphic to $\bigoplus_{i\in ...
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1answer
57 views

Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
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2answers
69 views

Free finitely generated modules

Let $A$ be a ring and consider the free modules $A^{\oplus n}$, $A^{\oplus k}$, with $n,k\in \mathbb{N}$. Can $A^{\oplus n}$ be isomorphic to $A^{\oplus k}$ if $k\neq n$? Thanks in advance for the ...
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1answer
57 views

Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
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4answers
170 views

How to show $\mathbf{Q} $ is not free

We know that torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show $\mathbf{Q}$ is not finitely generated and not free?
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1answer
35 views

When is a homomorphism an epimorphism?!

I want to prove the following characterization of an $R$-module homomorphism $g$ to be surjective: "whenever the composition of $g:M→N$ and $k:N→Y$ is zero, then $k=0$". It is easy to go one side: ...
4
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1answer
24 views

$M$ noetherian, $f$ endomorphism of $M$, $\operatorname{coker}f$ has finite length, then $\operatorname{coker}f^n$ and $\ker f^n$ have finite length.

Let $M$ be noetherian and let $f$ be an endomorphism of $M$. Suppose that $\operatorname{coker}f$ has finite length. Prove that both $\operatorname{coker}f^n$ and $\ker f^n$ have finite length ...
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1answer
23 views

Free module over a ring with identity with a basis of size m, ∀m≥n

Please, help on this Exercise [Hungerford's Algebra, IV.2.12] If $F$ is a free module over a ring with identity such that $F$ has a basis of finite cardinality $n\geq 1$ and another basis of ...
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1answer
155 views

Is this particular module flat?

Let $A=k[x^2,xy,y^2]\hookrightarrow B=k[x,y]$, where $k$ is a field. Is $B$ flat over $A$? I am guessing the answer is no. My first thought is, since $B$ is integral over $A$, so it's finitely ...
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2answers
73 views

On a proof that left artinian implies left noetherian

Questions: [Refer to below] Could one elaborate on $\rm\color{#c00}{(a)}$, $\rm\color{#c00}{(b)}$ and $\rm\color{#c00}{(c)}$ ? My thoughts : $\rm\color{#c00}{(a)}$ For $r+J\in R/J$ and ...
3
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1answer
25 views

bilinear maps with respect to noncommutative rings

Consider a noncommutative ring with unity $R$, three left $R$-modules $M,N,P$ and a map $f\colon\;M\times N\to P$ such that: $ f(m+m',n)=f(m,n)+f(m',n)\\ f(m,n+n')=f(m,n)+f(m,n')\\ ...
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2answers
65 views

not free modules

I'm not sure if this result is true or not. Let $R$ be a commutative ring and M a left $R$-module, if P is not free as a sub-module of M, can we say that M is not free either? thank you for your time
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1answer
44 views

Picard group of $\mathbb Z[\sqrt{-5}]$

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of ...
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1answer
23 views

Isomorphism between modules over a semisimple ring

If $P$ is a module over the semisimple ring $R/J$, where $R$ is a semilocal ring having $1$, and $J$ is its Jacobson radical, does any isomorphism $P⊕...⊕P≅P'⊕...⊕P'$ with the same (finite) number of ...
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0answers
15 views

From progenerators to progenerators

I know that if $R$ is a ring (with identity) and $P$ is a progenarator right $R$-module (a f.g. projective generator) then $P/PJ$ is clearly f.g. when $J$ is the Jacobson radical of $R$; but, how ...
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1answer
47 views

Are finitely presentable modules closed under extensions?

If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable? The answer is "yes" if we replace modules with groups, ...
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1answer
63 views

Given a, b How many solutions exists for x, such that: $a \bmod{x}=b $

Given $a, b$. How many solutions exists for $x$, such that: $$a \bmod{x}=b $$ By example: $a = 21$ and $b = 5$ $21 \bmod{8} = 21 \bmod{16} = 5$ Then $x$ has 2 solutions
7
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2answers
72 views

Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement ...
1
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1answer
22 views

Prove that if $M$ is an $R-$ projective left module then $M/IM$ is an $R/I-$ projective left module. [duplicate]

Let $I$ ba a two-sided ideal of a ring $R$ and $M$ be an $R-$ left module. Prove that if $M$ is an $R-$ projective left module then $M/IM$ is an $R/I-$ projective left module. It is easy to see that ...