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

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3answers
175 views

$IM$ is a submodule of M?

I would like to know if $IM$ is a submodule of $M$, I'm trying to use the submodule criterion which is: $G$ is a submodule of $M$ iff $G\neq \varnothing$ and whenever $g,g'\in G$ and $r,r'\in ...
3
votes
0answers
49 views
+200

Recovering free modules from their projective limit

Let $\dotsc A_2 \to A_1 \to A_0$ be a sequence of surjective homomorphisms of commutative rings. Consider the projective limit $\varprojlim_i A_i$. If $S$ is an (infinite) set, then $\varprojlim_i ...
0
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1answer
21 views

Complement but not direct summand

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
0
votes
0answers
8 views

Example of syzygies not arising from polynomial rings

The Wikipedia page on syzygies describes a syzygy in general as a relation between the generators of a module. However, I find it difficult to find examples of syzygies that does not involve ...
7
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2answers
103 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
0
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1answer
13 views

End(V) and End(V)xEnd(V) are isomorphic

Let R=End(V) be the ring of all linear endomorphisms of an infinite dimension complex vector space V with countable basis $\{e_{1},e_{2},...\}$ . Prove that R and RxR are isomorphic as left R-modules. ...
0
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1answer
33 views

How to show $\alpha=i_L$ and other equalities without using isomorphism?

This is a question of Commutative Algebra.It was given in my class.I was able to solve it to some extend.But I have some doubts. Please help me. Thnx in advance. $A$ is a commutative ring with ...
1
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1answer
64 views

Intersection of all associated primes

Given $(R,m)$, a Noetherian local ring, and $M$ a nonzero $R$-module. I was wondering if there is a way to describe the elements of $\displaystyle\bigcap_{P\in Ass_RM} P$. In particular, when $M$ is ...
2
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1answer
32 views

Localization of a finitely generated module is trivial iff its annihilator is nontrivial

I have a problem on Atiyah and MacDonald's commutative algebra book, the exercise 3.1: Let $S$ be a multiplicatively closed subset of a ring $A$ and $M$ a finitely generated $A$ - module. ...
0
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0answers
17 views

Relatively Projective/Relatively Injective modules which are not Projective/Injective

$G$ is a group, $\Lambda=\mathbb{Z}[G]$ is a group ring Induced Module is a $\Lambda$ module which is of the form $\Lambda\otimes_\mathbb{Z} X$ for some abelian group $X$. Coinduced Module is a ...
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1answer
22 views

A uniform module as an intersection

Let $R$ be a semiprime, nonsingular ring with finite Goldie dimension u.dim $R_R$. (Nonsingularity means here that $Z(R_R)=0$, where $Z(R_R)$ is the set of elements $x$ of $R$ with $ann(x)$ is ...
1
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0answers
32 views

Reference about Tensor Product

do you know some reference about tensor product of modules, with all elementary properties are proved ?? I want something a bit more explicit than "Commutative Algebra" of Atiyah and Macdonald. ...
1
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0answers
8 views

Does projectivity over the base ring imply projectivity over the extension?

Suppose $R\to S$ is a ring homomorphism. Let $M$ be an $S$-module. Suppose $M$ is projective as an $R$-module. Is $M$ projective as an $S$-module? If not, what additional hypothesis would suffice (for ...
2
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0answers
24 views

on basic theory over $\mathbb{Z}_4$-linear codes

I am a bit confused about this example in Huffman and Pless's Fundamentals in Error-Correcting Codes. This can be found in Chapter 12: Let $\mathcal{C}$ be a nonempty subset of $\mathbb{Z}{}^4_4$ ...
0
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1answer
21 views

Characterization of Zero in a module

Let $M$ be an $R-module$ over a ring $R$. Obviously, $0 \cdot m = 0, \forall m \in M$. Is it also true that $x \cdot m = 0, \forall m \in M$ if and only if $x = 0$?
0
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2answers
39 views

The difference between the free $A$-module $A^{(M\times N)}$, and $M\times N$

Let $M$ and $N$ be two $A$-modules, where $A$ is a commutative ring. What is the difference between the free $A$-module $A^{(M\times N)}$, and $M\times N$? In Atiyah-Macdonald, both of these seem to ...
16
votes
5answers
2k views

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell ...
3
votes
2answers
250 views

$C \otimes A \cong C \otimes B$ does not imply $A \cong B$

Let $R$ be a commutative unital ring and let $M$ be an $R$-module and let $S$ be a multiplicative subset of $R$. Today I proved both of the following: $$ S^{-1} R\otimes_R S^{-1}M \cong S^{-1} M$$ ...
2
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1answer
27 views

If $I$ and $J$ are distinct maximal ideals of $R$, any module homomorphism $R/I\to R/J$ is zero.

I am not sure how to deal with the following problem: $I$ and $J$ are distinct maximal ideals in a commutative ring $R$ and if $\phi$ is a modules homomorphism from $R$-module $R/I$ to $R$-module ...
0
votes
1answer
36 views

Proof of $0\rightarrow A^G\stackrel{f}\rightarrow B^G\stackrel{h}\rightarrow C^G\rightarrow 0$

When we have a $G$-modules exact sequence $0\rightarrow A\stackrel{f}\rightarrow B\stackrel{h}\rightarrow C\rightarrow 0$ we have a $G$-modules exact sequence $0\rightarrow ...
4
votes
3answers
83 views

Why do we define modules to be over *abelian* groups?

We define a module to be an abelian group $M$ together with a ring action $R \times M \to M$ that satisfies certain properties. Q: Why do we require that $M$ is abelian? I know that modules ...
0
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0answers
48 views

Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
0
votes
2answers
51 views

tensor an exact sequence

Let $R$ be a ring, $M$ an $R$-module, and $I$ an ideal of $R$. Then given the exact sequence: $$0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow0$$ when is $M$ flat. By some theorem, $M$ is ...
1
vote
2answers
41 views

Show that the homomorphism $\lambda: k[X] \to End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ has a nontrivial kernel.

$\DeclareMathOperator{\End}{End}$ I'm trying to show that: Show that the homomorphism $\lambda: k[X] \to \End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ as in (see ...
12
votes
2answers
1k views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
0
votes
1answer
32 views

Can the integral group ring construction be extended to groupoids in such a way that it provides a functor from Groupoids to Rings?

If $G$ is a group, and $R$ is a ring, one can construct the group ring $R[G]$, which is a free $R$-module with basis $G$, and with multiplication coming from the original multiplication of $G$. ...
1
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1answer
247 views

The relationship of free, divisible, projective, injective, and flat modules.

In general, we have that: free $\Rightarrow$ projective $\Rightarrow$ flat injective $\Rightarrow$ divisible ( ($\Rightarrow$) be ($\Leftrightarrow$) in PIDs) Simple Counter-examples: projective ...
1
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0answers
24 views

What maps of $k$-algebras $A\to B$ induce finite maps $\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$?

Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map ...
5
votes
2answers
68 views

Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
1
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0answers
20 views

Left and right module on the cohomology of a sheaf

Let $X$ a topological space, say a complex variety, and $\mathbb{C}_X$ its constant sheaf. $\mathcal{D}(X)$ is the derived category of sheaves of $\mathbb{C}_X$-modules. Let $F^\bullet\in ...
0
votes
1answer
65 views

Proof of the theorem: Any $R$-module is contained in an $R$-injective module

Note that this an exercise in Foote and Dummit's Algebra. Problem: Let $1\in R$. Any $R$-module $M$ is contained in an $R$-injective module. Definition : $R$-module $Q$ is injective if ...
1
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0answers
36 views

Proof of ${\rm Tor}\ (A,B)={\rm Tor}\ (B,A)$

Problem : I want to prove that ${\rm Tor}$ is symmetric where $1\in R$ Definition : Note that if $B$ has a projective resolution $$ P_n\rightarrow_{d_n}\cdots P_0\rightarrow B\rightarrow 0 $$ By ...
8
votes
1answer
114 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
3
votes
1answer
222 views

Nontrivial example of $M$ such that $\text{Ass}(M)=\varnothing$?

Suppose $R$ is a commutative ring, and $M$ an $R$-module. Is there a nontrivial example of such $M$ where the set of associated primes $\text{Ass}(M)=\varnothing$? Taking $M=0$ feels kind of ...
1
vote
1answer
49 views

Pushout,modules,short exact sequences

if we have this diagram in modules $$\begin{array}{ccccccccc} & & 0 & & 0 & & & & \\ & & \downarrow & & \downarrow & & & & \\ 0 ...
0
votes
1answer
76 views

Does every free $R$-module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
6
votes
0answers
61 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
1
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1answer
49 views

Failure of the Krull-Schmidt Theorem?

Theorem 1.19 of Representation Theory of Finite Groups: Algebra and Arithmetic is the Krull-Schmidt theorem, which I screenshotted and uploaded it here, I don't have any problem with this theorem and ...
2
votes
0answers
57 views

Help about trace of a module

I'm reading about Trace and Reject, which definition: $Tr_M(U)= \sum \lbrace Im(h)|h\in Hom_R(U,M) \rbrace$ $Rej_M(U)= \bigcap \lbrace Ker(h)|h \in Hom_R(M,U)\rbrace$ So is that right: if ...
1
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1answer
28 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
0
votes
1answer
36 views

A direct limit of pullbacks

Let an $R$-module $C$ be a direct limit of finitely presented $R$-modules $C_i$, and we have a short exact sequence as follows: $$0→A↪B\stackrel{\pi}→C→0.\qquad (S)$$ From each $C_i$ to the direct ...
-1
votes
3answers
67 views

Annihilator of annihilator of annihilator of a submodule

Let $R$ be a commutative ring and M be an $R$-module. The question is whether for every submodule $N$ of $M$ the equality ...
0
votes
0answers
43 views

Bass numbers of minimax modules are finite?

Let $R$ be a commutative Noetherian ring, and $M$ be a minimax $R$-module. Are the Bass numbers of $M$ are finite? (An $R$-module $M$ is called minimax, if there is a finite submodule $N$ of $M$, such ...
5
votes
2answers
130 views

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalance does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
1
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1answer
47 views

Example of a module such that every proper submodule is finitely generated but the module is not.

Let $R$ be a ring with 1 and $M$ an $R$-module. What is an example such that $M$ is infinitely generated but every proper submodule is finitely generated.
0
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1answer
75 views

Modules with finite support in $\mathrm{Max}(R)$

Let $R$ be a commutative Noetherian ring, and $M$ be an $R$-module. Is the following statement true? If $\mathrm{Supp}_R(M)$ (support of $M$) is a finite subset of $\mathrm{Max}(R)$ (the set of all ...
1
vote
1answer
34 views

Existence of a maximal submodule

Let $R$ be a left Artinian ring and let $M$ be a nonzero left $R$-module. Prove that $M$ has at least one maximal submodule. I don't really know where to start. Any hint? Thanks!
2
votes
2answers
23 views

A direct limit concerning some homomorphisms

In an algebra text there is the following argument I am stuck in the last part of which: "Let $f:B→C$ be an epimorphism in the category of $R$-modules, and $D=∑_{n=1}^∞c_nR$ be a countably generated ...
3
votes
1answer
38 views

Injective hull of $\mathbb{ Z}_n$ [duplicate]

What is the injective hull of $\mathbb Z_n$? I know that in case $n=p$ is prime, the injective hull would be isomorphic to $\mathbb Z_{p^∞}$, but in general case, I have no idea. Can anyone be of ...
1
vote
2answers
33 views

Cardinality of minimal generating set of a module is constant

Let $R$ be a commutative ring with unity and $M$ be a finitely presented module over $R$. Then how to show that for any minimal generating set $S$, the cardinality is same? Edit: Thanks to Martin to ...