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

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6
votes
1answer
464 views

question about typical proof of Krull Intersection Theorem

In Atiyah-Macdonald, in the proof of Theorem 10.17 (Krull's intersection theorem), the authors go through a 4-line chain of arguments to show that that the kernel $E=\bigcap_{n=1}^{\infty}\mathfrak ...
3
votes
2answers
133 views

Sub-modules of free modules

I'm going back through basic module theory notes, and I've come across a paragraph explaining that a sub-module of a finitely generated free module may not itself be free. In my course a free module ...
1
vote
2answers
261 views

Show that Q, as a Z-module, is a direct summand in a direct product of copies of Q/Z.

Prove that $\Bbb Q$, as a $\Bbb Z$-module, is a direct summand in a direct product of copies of $\Bbb Q/\Bbb Z$. This is a problem from Hilton and Stammbach's Homological Algebra. If this is ...
0
votes
0answers
71 views

Tensoring over a direct sum of R

Is tensoring over a ring R same as tensoring over a finite direct sum of R. I mean is $A \otimes_R B$ isomorphic to $A \otimes_{R^n} B$? And if it doesnt hold for any ring $R$, Does it hold for ...
3
votes
1answer
95 views

Why we can consider both modules as modules over $R_{(p)}$? (Bruns and Herzog, Theorem 1.5.9)

I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over $R_{(p)}$ in this theorem? ... i know that ...
3
votes
2answers
1k views

The definition “module of finite type”.

I know the definition of "module of finite type" Is that different from finitely generated module ? Thanks.
1
vote
2answers
167 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
1
vote
1answer
204 views

Show that the image or the kernel are submodule of R-module.

Let $R$ be Commutative ring and $M$ be an $R$-module. Show that $im(H)$ or $ker(H)$ are submodule of $R$-module $M$, where $$H\in Hom(M,-)$$ First, I think by lemma: If $M$ is an $R$-module and $N$ is ...
-1
votes
1answer
61 views

ring and module problem

Let $$F=\mathbb{R}$$ $$V=\mathbb{R}^{4}$$ consider two matrices $$S1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 ...
1
vote
0answers
91 views

Tensor product of two simple modules [duplicate]

Let $k$ be a field of arbitrary characteristic. Suppose that $A$ and $B$ are finite dimensional $k$-algebras. Let $S$ be a finite dimensional simple $A$-module and let $T$ be a finite dimensional ...
2
votes
1answer
72 views

equivalence in category

First gives some definitions, and then the property that I am confused. $A$, $B$ are both $R$-module, and $C$, $D$ an (additive) abelian group, consider the category $M(A,B)$ whose objects are all ...
2
votes
0answers
104 views

If $\alpha$ and $\beta$ are algebraic integers then the roots of $x^2+\alpha x+\beta$ are algebraic integers

(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.) I'm trying to solve this problem with ...
1
vote
1answer
104 views

How to show Not a Free Module

Let $\mathbb K$ be a field, $A= \mathbb K [x,y]$ and $ M = Ax + Ay$. prove that $M$ is NOT a free module!
1
vote
1answer
115 views

$\operatorname{Hom}(R\times R,M)$ isomorphic to $\operatorname{Hom}(R,M) \times\operatorname{ Hom}(R,M)$

If $M$ is a right $R$-module, then $\operatorname{Hom}(R \times R,M)$ is isomorphic to $\operatorname{Hom}(R,M) \times \operatorname{Hom}(R,M)$ My questions are: What is the isomorphism map? (I ...
0
votes
0answers
47 views

Divisible Direct Sum or Direct Product

We know that direct sum and direct product of divisible R-modules are divisible when R is a domain. Does there exist non-divisible modules with their product or sum being divisible? Or, if R is not a ...
2
votes
1answer
93 views

About the definition of the tensor product of modules

I was reading something about tensor product (balanced product) of modules (over an arbitrary ring $R$), but I cannot realize why we need a left and a right module. Would it be the same with two right ...
2
votes
1answer
189 views

Localization of a module as direct limit

Let $A$ be a commutative ring, $S \subset A $ a multiplicatively closed set and $M$ an $A$-module. For every $s \in S$ we denote by $M_{s}$ the localization of $M$ with respect to $\{ 1, s, s^2, ...
1
vote
0answers
67 views

Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
2
votes
0answers
105 views

Is the length of the composition series of a free module identical to the number of its bases?

Let $A_0$ be an Artinian ring, $M$ a free $A_0$-module. Then, is the length of the composition series of $M$ identical to the number of its bases? It seems to me that it is not. If $\mathfrak a$ ...
2
votes
2answers
73 views

Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...
1
vote
0answers
28 views

Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
5
votes
1answer
180 views

Can a ring isomorphism change the structure of a module?

Let $M$ be an $R$-module, where $R$ is a ring with unit. Given a ring automorphism $\phi: R \rightarrow R$, we can define a new $R$-module structure on $M$ by $r \cdot x = \phi(r) x$ for all $r \in ...
2
votes
1answer
61 views

A Sort of Exact Sequence

I have not given a lot of thought to this question: It may be very easy or very hard or somewhere in between. Suppose we have a sequence of modules and morphisms which looks like $ \ldots \to A_1 ...
3
votes
2answers
83 views

How can I compute $\operatorname{Tor}(\mathbb Z_{p},\mathbb Z_{q})$?

I am self-studying Vick's Homology Theory, and now it is on the topic of free resolutions. Since I am not familiar with it, I have little ideas about how to compute $$\operatorname{Tor}(\mathbb ...
1
vote
2answers
288 views

Free $A$-module isomorphic to a direct sum of copies of $A$?

Does this proposition hold even if it's not finitely generated? I think it does, since $M$ isomorphic to the direct sum of $M_i$, $M_i$ isomorphic to the direct sum of ...
2
votes
0answers
40 views

Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
0
votes
1answer
36 views

Annihilation and localization

Can somebody help proofing the following lemma. Let $x$ be an element of a module $M$, and let $\mathfrak{a}$ be its annihilator. Let $\mathfrak{p}$ be a prime ideal of $A$. Then $(Ax)_{\mathfrak{p}} ...
1
vote
1answer
63 views

Free modules and ideals

I am trying to show that an ideal I of R=$\mathbb{C}[x_1,x_2]$ generated by $x_1, x_2$ is free R-module. I am trying to show that I has a basis of the two generators given above. But I am not able to ...
0
votes
0answers
61 views

projective dimension of modules over triangular matrix rings

Let $R$ and $S$ be rings and $M$ be an $S$-$R$- bimodule. Then consider the triangular matrix ring constructed from these data (as, e.g., in Auslander-Reiten-Smalo). what one can say about the ...
2
votes
0answers
172 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
2
votes
1answer
113 views

What is the difference between a module of finite rank and finitely generated module.

R is an integral domain and every module we talk about is an R-module. If a module is finitely generated then obviously every element of the module can be written as finite R-linear combination of the ...
0
votes
1answer
100 views

The meaning of a symbol in the proposition

The question is what's the meaning of the symbol $\phi$? If it just a mapping,what's the mean of the equation? I guess it's $\phi(x)$, $x$ is the element of $M$. Then $\phi(x)$ is the element of ...
1
vote
1answer
68 views

help in an example.

In page 226 of David Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry there is an example which I need help in some parts of it: why $codim I= 1$? why $dim M = dim R = ...
3
votes
1answer
49 views

Show polynomials $I$ is not finitely generated as $R$-module

Let $R=\{a_0+a_1X+\cdots+a_nX^n\;|\;a_0\in\mathbb{Z},a_1,a_2,\dots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}_{\geq 0}\}$ and $I=\{a_1X+\cdots+a_nX^n\;|\;a_1,a_2,\dots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}^+\}$. ...
0
votes
1answer
42 views

Prove module $M$ is finitely generated if $N$ and $M/N$ are finitely generated

Let $R$ be a ring with $1$ and $N$ be a submodule of an $R$-module $M$. Prove that $M$ is finitely generated if $N$ and $M/N$ are finitely generated. There are two definitions of "finitely ...
1
vote
2answers
91 views

If $N$ and $M/N$ are free modules of finite rank, so is $M$

Definiton: A $R$-module $M$ is said to be free of finite rank if $M\cong R^k = R\times R\times\cdots \times R$ (k times). Let $R$ be a ring with $1$ and $N$ be a submodule of an $R$-module $M$. ...
2
votes
1answer
100 views

Is any right exact sequence of modules induced by free modules?

Let $R$ be a ring and let $M \to N \to K \to 0$ be an exact sequence of $R$-modules. Is there an exact sequence of free modules $A \to B \to C \to 0$ and a commutative diagram $$\begin{array}{c} M ...
0
votes
1answer
48 views

Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
1
vote
1answer
60 views

Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
0
votes
1answer
106 views

Span and Smallest Submodule Proof

Let R be a ring, M a R-module, and $X \subseteq M$ Show that span$(X)$ is the smallest submodule of R containing X. My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and ...
3
votes
1answer
41 views

$\operatorname{rank}(F) = \operatorname{dim}_{k}(\frac{F}{mF})$

Let $R$ be a commutative ring with unit; $m$ is a maximal ideal; $F$ a free $R$-module. We know that $\frac{F}{mF}$ is a vector space over $\frac{R}{m} = k$ . I have to prove that ...
19
votes
2answers
488 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
0
votes
2answers
276 views

Cardinalities of bases of a free $R$ module are same? [duplicate]

Let $R$ be a ring with no zero divisiors such that for all $r,s\in R$ there exist $a,b\in R$ not both zero with $ar+bs=0$. If $R=K\oplus L$ then $K=0$ or $L=0$. if $R$ has an idendity then any two ...
0
votes
1answer
185 views

Are A and Hom(A,Q/Z) isomorphic?

If A is a Z-module, does this imply that A and Hom(A,Q/Z) are isomorphic ?
0
votes
2answers
394 views

Possible difference between $\mathbb{Z}$-modules and vector spaces

Suppose $G$ is a free abelian groups, i.e. a free $\mathbb{Z}$-module; we have a set $S \subset G $ such that $S$ spans $G$. Can we conclude that the rank of $G$ as a $\mathbb{Z}$-module is $ \leq ...
3
votes
1answer
62 views

A gentle reference for flat modules with exercises

I would like to learn about sources that give a gentle introduction to flat modules, with exercises. Could you recommend me some source? This might be a section in a book, or some article. I am ...
2
votes
1answer
95 views

Relation Between Free Quotients and Modules

Here is my question: Let $M$ and $M'$ be $R$-modules, where $R$ is a commutative ring, and $N \subseteq M$ and $N' \subseteq M'$ submodules. Suppose that $N \cong N'$ and $M/N \cong M'/N'$. Determine ...
2
votes
1answer
443 views

Definition/existence/uniqueness of a minimal projective resolution

I'm reading Dave Benson's book "Representations and Cohomology," Volume I, and I'm trying to understand the following discussion on page $32$ in which he introduces the notion of a minimal projective ...
4
votes
1answer
89 views

Don't understand a proposition and its proof

Proposition 5.1 from Commutative Algebra by Atiyah and Macdonald: $x∈B$ is integral over $A$,then $A[x]$ is a finitely generated $A$-module. The elements in $A[x]$ are the set of all the sum. If ...
2
votes
1answer
338 views

Understanding the Bockstein homomorphism in group cohomology

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism ...