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

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16 views

Different definitions of submodule

I have stumbled upon two different definitions of a submodule and I cant see why they are equivalent, let me state them ($M$ is the module and $A$ is the commutative ring with unity): First ...
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1answer
44 views

Does a module always have a cyclic non-zero quotient?

Let $M$ be a module over a ring $R$. My question is: Does there exists a submodule $N$ such that $M/N$ is isomorphic to $Ra$ for some $a\in M\setminus N$, that is, $M/N$ is 1-'dimensional'?
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1answer
23 views

$\text{Coker}(f),\text{Coker}(g)$ projective modules $\Rightarrow$ $\text{Coker}(gf)$ projective

Let $\textbf{Ch}_R$ the category of chain complexes of $R-$modules ($R$ is an associative ring with unit). I want to prove that this cat. satisfies the model category axioms. In particular we want to ...
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43 views

On semisimple rings

Let $R$ denote a ring with unity. I know that, if $R$ is semisimple, then every $R$-module is semisimple. In particular the class of indecomposable $R$-modules coincides with the class of simple ...
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1answer
58 views

Atiyah-MacDonald, 3.18. Why is $B_q$ a local ring of $B_p$?

This question is on the hint that the book gives to finish the exercise. Namely, if $f: A \rightarrow B$ a flat homomorphism of rings, $q$ a prime ideal of $B$ and $p = q^c$, then $B_q$ is a local ...
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11 views

Can the injective module map be seen as the composite of the injective maps like this?

Here A,B are R-modules, and the map f:A to B is injective. I wonder if I can see the map f as the composite of injective module maps $f_1f_2...f_n$, such that $f_1$ is the injective map from A to ...
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32 views

Specific examples of submodules.

I have two issues for which I seek solutions for a long time : 1- Submodules of finitely generated modules need not be finitely generated. I searched and i found that the submodule $K$ of the ...
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1answer
30 views

about homomorphismes of modules

My issue is the following: Let $M$ be an $A$-module with $A$ a commutative unitary ring, $u: M\longrightarrow M $ an $A$-morphism. I cannot solve or find the proof of the following assertions: 1- ...
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1answer
44 views

Properties of module length

Let $e_{A}(\phi, M): = l_A(\mathrm{coker}(\phi) ) - l_A(\ker(\phi))$. In my book it is stated that if $IM = 0 \implies e_{A}(\phi, M) = e_{A/I}(\phi, M)$ and this seems to be obvious for the author. ...
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3answers
153 views

Proving that two quotient modules are isomorphic

Given a ring $R$ and $R$-modules $A,B,C,D$ such that $$\sigma:A \rightarrow B, \tau: C \rightarrow D, \rho: A \rightarrow C, \kappa: B \rightarrow D, \ \mathrm{and} \ \kappa \circ \sigma = \tau \circ ...
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1answer
44 views

Prove that $Ker(g \otimes k)= Im(f \otimes 1_{N}) + Im (1_{M} \otimes h)$

Suppose we have two short exact sequences: $$0 \to M' \mathrel{\overset{f}{\to}} M \mathrel{\overset{g}{\to}} M'' \to 0 $$ in Mod-R $$0 \to N' \mathrel{\overset{h}{\to}} N \mathrel{\overset{k}{\to}} ...
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1answer
38 views

Proving that a module can be decomposed as a direct sum of submodules

Letting $X$ be a ring and $K$ be an $X$-module, I need to show that if $K \cong A \times B$ for some $X$-modules $A,B$, then $\exists$ submodules $M'$ and $N'$ of $K$ such that: $K=M' \oplus N'$ $M' ...
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29 views

Does a finite generated submodule with another element always generate a finite submodule?

Let M be a module and N be a finite generated submodule of M, and x$\in$M\N, then consider the existence of module B which is generated by N and x. I think, as N is f.g thus we can see N as ...
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1answer
45 views

Why is this Hilbert's Syzygy theorem?

In Lang's Algebra, chapter XXI, §4, on p. 861 he describes the standard construction of a graded (in principle infinite) free resolution of a finite graded module $M$ over the polynomial ring $A = ...
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32 views

If $Ext_A^n(M,N) \ne 0$, then $Ext_A^n(M,N') \ne 0$, for every indecomposable summand N' of N?

Let $A$ be an artin algebra and $M$ and $N$ finitely generated modules over $A$. Suppose that $Ext_A^n(M,N) \ne 0$, is it possible to conclude that $Ext_A^n(M,N') \ne 0$, for each indecomposable ...
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35 views

Is $\text{Hom}_R(A,B)$ isomorphically equal to $\text{Hom}_R(B,A)$?

This question is as in the title, where $R$ is a commutative ring and $A,B$ are $R$-modules. I have no references for that, I just thought of it by myself, somehow. I know this is true when $A,B$ are ...
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0answers
19 views

module theory and Hom Functor

Let $M$ and $N$ are two non-zero modules over $R$. Can we have Hom(N,M)=Hom(M,N)=0? Thank you. I'm stuck in a demonstration and I have to use this results
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1answer
59 views

Intuitive explanation of Four Lemma

In the Short Five Lemma where the rows are exact, it is a fact that $$\alpha \text{ and }\gamma \text{ injective (surjective) }\implies \beta \text{ injective (surjective)}.$$ I've heard this fact ...
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18 views

Prove that $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$

Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup ...
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1answer
43 views

(An arbitrary direct product of free modules need not be free)

For each positive integer $i$ let $M_i$ be the free $\Bbb Z$-module $\Bbb Z$, and let $M$ be the direct product $\prod _{i \in \Bbb Z^+} M_i$. Each element of $M$ can be written uniquely in the form ...
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1answer
22 views

Projective modules and enveloping algebra

This question have two parts. The first one: Let A be a k-algebra that is projective as k-module, where k is a commutative ring. why is $A\otimes_kA$ projective as $A^{ev}=A\otimes_kA^{op}$-module? ...
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1answer
26 views

Equivalent properties of projective modules

For a left $R$ module $P$, the following are equivalent: Given $M\xrightarrow{\psi} N \to 0$ exact and $\omega: P \to N$, there exists $\tilde{\omega}: P \to M$ such that $\psi \circ ...
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1answer
44 views

What would be an example of free module such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has the IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=\mathrm{rank}(M)$. Let ...
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0answers
14 views

Prove tensor multiplication is bilinear [duplicate]

I am reading these online notes(http://people.virginia.edu/~mve2x/7752_Spring2010/lecture5.pdf), and on the bottom of page 2 the following exercise occurs. Define tensor multiplication of two ...
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1answer
33 views

Basis for proper rational functions

Suppose $F$ is a field, and let $F(x)$ denote the $F$-vector space of all rational functions $\frac{f(x)}{g(x)}$, where $f,g\in F[x]$ are polynomials, with $g$ different from zero. Let $F(x)_p$ denote ...
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0answers
28 views

Schur and Weyl modules.

Let $m$ be a non-negative integer and $\lambda=(\lambda_1, \cdots, \lambda_s)$ a partition of $m$. If $V$ is a vector space of dimension $n$ (over a field $\mathbb{K}$), we can consider the Schur ...
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1answer
55 views

Under what conditions $M \cong (M/N) \oplus N$ where $M$ is an $R-$module and $N$ is a submodule?

I'm curious about it. I know that if $M=A\oplus B$ then $M/A \cong B$ (identifying $A$ with $A\oplus 0$), one would expect that the "converse" holds, i.e., if $M/A\cong B$ then $M\cong A\oplus B$. Is ...
2
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1answer
20 views

If f is surjective then f is not a right divisor of zero

Let $R$ be a ring and $M$ a R-module. For $r\in R$ define $f:M\to M$ by $f(s)=sr$. Show that $f$ is injective if and only if $r$ is not a right zero divisor. I have done a similar problem to this in ...
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1answer
33 views

Induced map of quotient modules

Suppose that $R$ is a ring, $I \subset R$ is an ideal, and $f : A \to B$ is a homomorphism of $R$-modules. Is it true that there is an induced homomorphism $f/I : A/IA \to B/IB$ which is a map of ...
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1answer
31 views

Clarification between a module and a vector space?

I'm reading Kenneth Hoffman's Linear Algebra, Ed2. In $\S5.5$ it talks about Module and Vector Spaces: (1) If $K$ is a commutative ring with identity, a module over $K$ ( or a $K$-module) is ...
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1answer
12 views

Equivalence of Semisimplicity

In Noncommutative algebra by Benson Farb, there is an exercise concerning this result; An R-module M is semisimple if every submodule of M is a direct summand. where semisimple is defined as M being ...
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1answer
21 views

How can I show that any one to one endomorphism of an Artinian module is an automorphism?

How can I show that any one to one endomorphism of an Artinian module $M$ is an automorphism? I was given this question and I presume that it is really to show that Artinian modules are co-hopfian. ...
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1answer
35 views

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$?

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$? Here $E(M)$ is the injective envelope of $M$ and $\operatorname{Ass}$ denotes the set ...
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17 views

Can matrix representation be extended to free modules?

Let $V$ be a finite-dimensional $F$-vector space and $T$ be an $F$-endomorphism on $V$. Let $\alpha$ be an $F$-basis for $V$. Then, there is a natural $F$-linear isomorphism ...
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15 views

equivalent to finitely cogenerated

During a lecture last week, my lecturer used this fact during a proof $_RM$ is finitely cogenerated if and only if for every family {$_RN_i : i \in I$} of submodules $_RN_i$ of $_RM$ such that ...
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1answer
14 views

Is the kernel of a map to a finitely presented module finitely generated?

Let $R$ be a ring (unital, not necessarily commutative), $M$ a finitely presented left $R$ module. Suppose $m_1,\ldots,m_n\in M$ generate $M$. This determines a surjection $f:R^n\to M$. Must the ...
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30 views

An essential right ideal in a ring

Let $S⊆R$ be rings with unity such that $S_S$ is essential in $R_S$. If $r∈R$ is a nonzero element there exists an $s_0∈S$ with $rs_0$ a nonzero element of $S$. Now, could we find a right ideal $I$ ...
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42 views

Proving that $\det(A)R^n \subset\mathrm{Im}(\Phi)$

Let $R$ be a commutative ring with unity and consider the free $R$-module $R^n$. Given a matrix $A$ with coefficients in $R$, define a homomorphism $\Phi: R^n\to R^n$ by $\Phi(u) = Au$. My question is ...
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1answer
23 views

$M' \to M \to M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \to \text{Hom}(M,N) \to \text{Hom}(M',N)$ is exact.

Let, $M', M'', M, N$ be $A$-modules. $M' \stackrel{u}{\to} M \stackrel{v}{\to} M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \stackrel{\bar{v}}{\to} \text{Hom}(M,N) \stackrel{\bar{u}}{\to} ...
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1answer
32 views

Is the symmetric algebra direct sum of $k$-th symmetric powers?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $S(M)$ be the symmetric algebra of $M$ and $S^n(M)$ be $n$-th symmetric power of $M$. Then is $S(M)$ the internal direct sum of $S^n(M)$? ...
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32 views

What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...
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79 views

Quotient of direct sum of $l$ copies of a PID by an ideal.

Let $R$ be a PID, and $\psi:R^k\to R^l$ an homomorphism. I would like to know under what circumstances it's true that: $$ R^l/\operatorname{Im}\psi \cong R/I_1\times \cdots \times R/I_l, $$ for some ...
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2answers
40 views

equivalent condition for finitely generated module

Let $_RM \in R$-Mod for a unital ring $R$. Show that the following two conditions on $_RM$ are equivalent (a) $_RM$ is finitely generated; (b) for every family $\{_RM_i : i \in I \} \subseteq R$-Mod ...
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1answer
8 views

Integral group ring quotient as a right module

Let $G$ be a group and let $N$ be normal in $G$. Consider the ring homomorphism of group rings $\mathbb{Z}G\to\mathbb{Z}(G/N)$ defined by $g \mapsto gN$. Denote its kernel by $I_N'$. Let $I$ be the ...
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1answer
11 views

Is it okay to define k-th symmetric power of $M$ in this way?

I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below. Let $R$ be a commutative ring and $M$ be an $R$-module. ...
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1answer
18 views

What condition is necessary to form a quotient algebra?

Let $R$ be a ring and $A$ be an $R$-algebra and $B\subset A$. What condition should $A,B,R$ have to get the concept of quotient $R$-algebra? So that we can define $(a+B)+(b+B)=(a+b)+B$ and ...
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18 views

What's actually $S^k(M)$?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $T(M)$ be the tensor algebra of $M$. Then, what is $S(M)$ (symmetric algebra and $S^k(M)$? Some articles define $S(M)$ as a quotient of ...
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1answer
37 views

Can every $T$-stable subspace be realised as the kernel of another linear operator that commutes with$~T$?

This is inspired by the question Is Every Invariant Subspace the Kernel of an polynomial applied in the operator?, where "invariant subspace" and "polynomial in" are relative to a given linear ...
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16 views

Under what condition, does this universal property of tensor algebra hold?

Let $R$ be a commutative ring and $M$ be an $R$-module. Note that the tensor algebra $T(M)$ is a unital associative $R$-algebra. Below is the universal property of the tensor algebra. Theorem: ...
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1answer
13 views

Why do we take $R$ when constructing the tensor algebra?l

Let $R$ be a commutative ring and $M$ be an $R$-module. Define $T^0(M)=R$ and $T^n(M)=M\otimes...\otimes M$(n-times) for $n\in\mathbb{Z}^+$. Then we take $T(M)\triangleq \oplus T^n(M)$ and give an ...