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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1answer
30 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 $Hom(eR,M)$ ...
2
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0answers
29 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 it's alternative ?
2
votes
1answer
54 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 ...
0
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0answers
26 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 ...
0
votes
1answer
34 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 ...
0
votes
2answers
43 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 ...
2
votes
1answer
60 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.
0
votes
1answer
16 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 ...
1
vote
1answer
50 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 $P$, ...
1
vote
1answer
24 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. ...
1
vote
1answer
46 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 ...
0
votes
2answers
35 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 $5$?
5
votes
3answers
64 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 ...
0
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1answer
55 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 $\mathsf{Mod}$-$R$?
4
votes
2answers
71 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 ...
1
vote
2answers
51 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
votes
2answers
46 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 $a_i,b_i\in\Bbb{Z}$. ...
0
votes
1answer
33 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
votes
1answer
37 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 ...
1
vote
1answer
50 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? ...
4
votes
2answers
67 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 ...
1
vote
1answer
53 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 $t(sb'-s'b) = 0$. ...
3
votes
4answers
165 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?
0
votes
1answer
31 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
votes
1answer
20 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 ...
0
votes
1answer
22 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 ...
5
votes
1answer
148 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 ...
1
vote
2answers
67 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
votes
1answer
23 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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votes
2answers
60 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
4
votes
1answer
41 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 ...
2
votes
1answer
22 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 ...
0
votes
0answers
14 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 ...
3
votes
1answer
43 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, ...
1
vote
1answer
60 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
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votes
2answers
58 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
vote
1answer
20 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 ...
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1answer
14 views

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $.

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $. Help me some hints. Thank you in advance.
2
votes
1answer
99 views
+50

Two properties related to semisimple rings

Let $R$ be a semisimple ring Show the following (i) If $xy=1 \in R$, then $yx=1$. (ii) If $x \in R$ is such that $xR$ is a left ideal of $R$, then $xR=Rx$. I am pretty lost with the two items. I ...
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vote
2answers
28 views

IM is a submodule of M

In the fisrt answer of this question where $R$ is a ring, $I$ is a left ideal of $R$ and $M$ an $R$-module; I don't know why can't I see that $IM$ is closed under addition. If we take two elements $x$ ...
1
vote
1answer
23 views

Equivalence of categorical coproduct proof

quiLet $C$ be an abelian category and {$X_1$,...,$X_n$} a finite family of objects in that category. ( $X$,($M_i$: $X_i$$\to$ $X$) where $i_1$=1,....n a coproduct of the finite family if and only if ...
0
votes
2answers
30 views

Prove that an exact sequence splits

Let $0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0$ where n =rs an exact sequence of $\mathbb Z$ modules the how can I prove the sequence splits if and only if $(r,s)=1$ The only thing ...
2
votes
1answer
21 views

Semisimplicity of the ring $\mathbb Z_n$

I am being asked to figure out when $\mathbb Z_n$ is a semisimple ring. It is clear to me that if $n$ is prime then $\mathbb Z_n$ is simple, which implies it is semisimple. If $n=p_1...p_n$ is a ...
0
votes
0answers
20 views

Simple problem about morphism in abelian categories

$f$ : $X\to$ $Y$ and $g$ : $Y\to$$Z$ a sequence in abelian categories. Show that if $gf$=$0$ if and only if exist a monomorphism $h$:$Im(f)$ $\to$ $Ker(g)$ such $kh$=$j$, where $j$:$Im(f)$$\to$ $Y$ ...
0
votes
1answer
24 views

Exact sequence of modules exercise

Show that if $$0 \rightarrow M_1 \xrightarrow{f} M_2 \xrightarrow{g} M_3$$ is an exact sequence of $R$-modules, then for all $R$-module $$0 \rightarrow \operatorname{Hom}_R(M,M_1) \xrightarrow{f_*} ...
3
votes
1answer
38 views

Finite abelian groups (application of structure theorem)

Problem Find all finite abelian groups that simultaneously have exactly $7$ elements of order $2$, exactly8 elements of order $3$, exactly $8$ elements of order $4$, at least an element of order ...
1
vote
1answer
52 views

Constructing pullback and pushout problem

i) Let $p$ be a prime and $f: \Bbb Z \rightarrow \Bbb Z_p$ and $g: \Bbb Z_{p^2}\rightarrow\Bbb Z_p$ be the canonical epimorphism. Show that the pullback of $f$ and $g$ is isomorphic to $\Bbb Z ...
3
votes
1answer
58 views

Are two bimodules isomomorphic as left and right modules also isomorphic as bimodules?

let R be a commutative ring, and M, N two bimodules over R, such that there exists f : M -> N an isomorphism of left R-modules, and g : M -> N an isomorphism of right R-modules. Then are M and N ...
1
vote
1answer
35 views

Exact sequence and Noetherian modules

Let $R$ be a ring, $X,Y,Z$ and $T$ four $R-$ modules such that there exists a short exact sequence $$0 \rightarrow X \xrightarrow{f_1} Y \xrightarrow{f_2} Z \xrightarrow{f_3} T \rightarrow 0 $$ Prove ...
3
votes
1answer
38 views

Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other. [duplicate]

Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other. I need some hints to prove it. Thank you very much.