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

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2
votes
1answer
31 views

Maps involving direct sums of modules

This might seem like a strange question... I am curious: If $A$ is a ring and $M_1, M_2, M_3, M_4$ are $A$-modules, and we have a map $f: M_1\oplus M_2 \rightarrow M_3\oplus M_4$, then is there some ...
5
votes
2answers
53 views

S,R bimodules subcategory of the category of S+R modules?

If $R$ and $S$ are commutative rings, then does the category $R \oplus S$-modules encompase the category of $(S,R)$-bimodules? I was thinking we can accomplish this by defining the action to be: ...
0
votes
1answer
26 views

Can I conclude that the following map is surjective?

I have a module homomorphism $A \rightarrow B \oplus C $ whose projection onto the first factor $B$ is surjective. If the projection onto $C$ is surjective and $C$ is a simple module, can I conclude ...
1
vote
1answer
30 views

Projective modules over a direct product of rings

Let $R$ and $S$ be rings, and let $\text{proj}(R)$ denote the category of finitely generated projective modules. Is there an equivalence of categories between $\text{proj}(R \times S)$ and ...
0
votes
2answers
34 views

How to solve this problem about faithful module.

Let $R$ be a non-zero commutative ring with identity and $M$ a unital $R$-module. The $R$-module $M$ is called faithful if $rM=0$ for $r\in R$ implies $r=0$. Let $M$ be a finitely generated ...
2
votes
1answer
24 views

If $\{v_0,v_1\}$ and $\{v_1,v_2\}$ form a basis for $\mathbb Z^2$ then $v_2=-v_0+a_1v_1$

I am reading the section on non-singular surfaces from Fulton's book, Introduction to Toric varieties. There we have the following - Let $v_0,v_1,\dots,v_d=v_0$ be a sequence of lattice points in ...
0
votes
2answers
32 views

Embedding $\mathbb{Z}$ into a $\mathbb{Q}$-module

$\mathbb{Z}$ is not a $\mathbb{Q}$-module since if $\frac{1}{2}\cdot1=x\in \mathbb{Z}$, then $2x=1$, which is absurd. However, according to Dummit and Foote (3rd Ed, page 359), there exists an ...
3
votes
0answers
87 views

Geometric interpretation of algebraic property

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, ...
0
votes
1answer
25 views

on the splitting of a presentation

Let $P$ be an $R-$module and let $F$ be a free module (say $F$ isomorphic to $R^n$). If $P$ is a summand of $F$, clearly there is a presentation (namely a surjective $R-$homomorphism) $p:F\to P$. ...
1
vote
1answer
68 views

Is an $R$-module equivalent to an $R[X]$-module?

This is probably a really simple question but I'm a little confused. I'm reading through a proof that concerns a finitely generated $R$-module $M$. The very first line is: Consider $M$ as an ...
2
votes
1answer
28 views

Defining a radical of a module axiomatically

In these notes by Richard Vale, he approaches the notion of a radical of a module axiomatically, by saying the following. A radical is an assignment, to each R-module $M$, of a submodule $\tau(M) ...
1
vote
1answer
30 views

Modulo Arithmetic - Find smallest divider greater or equal to.

I'm looking for a nice solution to the following problem: x mod(y + d) = 0. Where x is a positive integer and y is a positive integer smaller or equal to x. I'm looking to find the smallest d ...
0
votes
1answer
50 views

How does one compute quotients of $\Bbb Z$-modules?

Doing homology computations inevitably involves computing quotients of $\Bbb Z$-modules, and I am not familiar with any way of doing this. Some examples I have been working with are: $$\frac{\Bbb Z ...
0
votes
1answer
28 views

a question about finitely generated modules

Let $M$ be an $R-$module. Let $R^n$ be the direct sum of $n$ copies of $R$. Prove that if there is a surjective $R-$homomorphism $\pi:R^n\to M$ then $M$ is finitely generated. Clearly $R^n$ is ...
1
vote
1answer
30 views

Show that for a Noetherian ring R, the R-module R$^{n}$ is Noetherian

I'm allowed to use this: An R-module M is Noetherian when a submodule N is Noetherian and their quotient M/N is Noetherian. In my particular problem, M = R$^{n}$ and N = R. I believe I will have ...
1
vote
1answer
40 views

Baer Sum notation requires clearence.

I am working on Baer sum and I have my book by Rotman, Introduction to Homology, and also MacLanes book Homology and they use notation I am puzzled on. I have understood baer sum of extensions ...
2
votes
0answers
30 views

Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
0
votes
2answers
56 views

Subgroups of $\mathbb Z^5$

Is there a nice way to see that any subgroup of $\mathbb Z \times \mathbb Z \times \mathbb Z \times \mathbb Z \times \mathbb Z$ can have at most 5 generators? I know that $\mathbb Z$ is Noetherian, so ...
2
votes
0answers
41 views

If $E$ is a finitely presented left $A$-module, then tensor product by E commutes with direct product.

Please can anyone help me to prove this statement: If $E$ is a finitely presented left $A$-module, then for every family $(F_i)$ of right $A$-modules the canonical homomorphism $\phi: ...
3
votes
1answer
48 views

Baer sum of $\mathbb{Z}_9$ and $\mathbb{Z}_9$

I am working on trying to figure out the third extension of $\mathbb{Z}_3$ by $\mathbb{Z}_3$, I know one is $\mathbb{Z}_9$ and the neutral element (with respect to baer sum) $\mathbb{Z}_3\oplus ...
0
votes
1answer
13 views

$M/aM\rightarrow N/aN$ surjective implies $N/N'/a(N/N')$ is zero

I found the following statement in a problem sheet which I don't know how to approach. This is the following: If $u:M \rightarrow N$ is an $A$-module homomorphism and we denote $N'$ the image of $u$. ...
1
vote
2answers
36 views

Atiyah and Macdonald's proof of the existence of the tensor product

I have a question regarding the proof of the proposition 2.12 in Atiyah and Macdonald's book. They say: " Let C denote the free A-module $A^{(M \times N)}$. The elements of C are formal linear ...
1
vote
0answers
43 views

How to show localization commutes with finite products/arbitrary coproducts?

I'm trying to do the following exercise from Vakil's notes on Algebraic Geometry: 1.3.F. EXERCISE. Show that localization commutes with finite products, or equivalently, with finite direct ...
0
votes
0answers
19 views

Finite generation and associated graded modules

My question is as follows: Let $R$ be a ring and let $M$ a right $R$-module. Suppose that $(F_i)_{i \geqslant 0}$ is a filtration of $R$ and $(M_i)_{i \geqslant 0}$ is a compatible filtration of ...
1
vote
1answer
43 views

Irreducible Module in $\mathbb Q$ and $\mathbb C$ [closed]

Consider $V=(x,y,z)$ in $\mathbb{Q}^3$ and $\mathbb Q[X]$ a polynomial ring. Define $\mathbb Q[X]$-module by $X(x,y,z)=(-5z+5y,x+z,z)$. How can I show that V is irreducible $\mathbb Q[X]$-module? Now ...
4
votes
1answer
55 views

Direct limit of completions of finitely generated submodules

Let $A$ be a noetherian, local, integral domain with maximal ideal $\mathfrak m$. Moreover let $M$ be an $A$-module; I'd like to know if there exists an explicit expression of the module: ...
1
vote
2answers
34 views

Show that $M$ is not maximal in $\mathbb Q$

$\mathbb Q$ is a $\mathbb Z$ module. Let $M$ be a proper submodule of $\mathbb Q$. Show that there is some $0 \neq b \in \mathbb Z$, such that $1/b \notin M$ If for all $b \in Z$, $1/b \in M$, ...
2
votes
1answer
20 views

A generalisation of fiber sums of bimodules?

It would be good to draw a diagram here, but as far as I understand, it's impossible. This resembles the notion of fiber sum, so I hope it is well-known. Let $A$ and $B$ be (unital) algebras over ...
0
votes
1answer
68 views

Decomposition of Free Module over PID

Let $M$ be a free module with rank $2$ over the PID $R$ having basis $B=\{b_1,b_2\}$. I have a few questions regarding the submodule $Rm$, where $m$ is some element of our module $M$. Suppose $m$ is ...
4
votes
1answer
28 views

Find the direct summands of $\mathbb Z/ 36\mathbb Z$

I have this exercise: Find the direct summands of the $\mathbb Z$-module $M = \mathbb Z/36 \mathbb Z$. If $\bar T$ and $\bar N$ are direct summands of $M$ then $\bar T \cap \bar N = \{\bar 0\} = ...
4
votes
3answers
49 views

Submodules of a product of simple modules

Let $R$ be a ring and $A,B$ two simple $R$-modules. I would like to prove the following: If $A$ and $B$ are not isomorphic, then the only submodules of $A \times B$ are $\{0\} \times \{0\},A ...
2
votes
1answer
26 views

Under what conditions is $J\cdot M$ an $R$-submodule of $M$?

I have that $M$ is an $R$-module where $R$ is commutative and unitary ring. Supposing that $J$ is an ideal of $R$, when is the set $J \cdot M$ an $R$-submodule of $M$? I have to check the two axioms ...
0
votes
2answers
29 views

$\mathbb Q$ as a $\mathbb Z$-module is not finitely generated

$\mathbb Q$ is obviously a $\mathbb Z$-module, however, it is not finitely generated. I can't figure out why. If $\mathbb Q$ is finitely generated, then there are $x_i \in \mathbb Q$ such that ...
0
votes
1answer
33 views

The presentation matrix $A$ of an $R$-module is the same as deleting a column of zeros of $A$

This is a proposition from Artin's Algebra (1st edition, Proposition 5.12): The presentation matrix $A$ of an $R$-module is the same as deleting a column of zeros of $A$. He proves it by ...
3
votes
3answers
75 views

What is the definition of direct sum of submodules?

Given a ring $R$ and $M_1,\ldots,M_n$ $R$-submodules of an $R$-module $M$, what is the definition of this set? $$\bigoplus_{i=1}^n M_i$$ From where I am reading it seems that it is: $M_1 + \cdots + ...
2
votes
1answer
32 views

Find an exact sequence from a module $M$ to direct sum of localizations of $M$ in a finitely generated ring

Suppose $f_1,f_2,\ldots, f_k\in R $ such that $R=(f_1,f_2,\ldots, f_k)$. Let $M$ be an $R$-module. Define $U(f)=(f,f^2,f^3,\ldots)$ and let $$M_i=U(f_i)^{-1}M$$ $$M_{i,j}=U(f_if_j)^{-1}M$$ ...
0
votes
1answer
13 views

$\text{Hom}(M \otimes_A N, L) \approx \mathscr{L}(M,N; L)$ The $A$-linear homs from the tensor product into $L$ are isomorphic with bilinear maps.

Let $M,N, L$ be two $A$-modules over a commutative ring $A$. Let $\mathscr{L}(M,N;L)$ be the $A$-module of bilinear maps $M \times N \to L$. Then $\text{Hom}_A(M \otimes_A N, L) \approx ...
0
votes
0answers
23 views

Doubt about the module of coinvariants

Suppose that $G$ is group, $R$ is a ring (commutative, associative with $1\neq0$) and $M$ is a left $RG-$módulo such that $M$ is free as a left $R-$module. Is it true that the module of coinvariants ...
1
vote
3answers
47 views

Cyclic Modules, Characteristic Polynomial and Minimal Polynomial

Suppose that $\mathrm{dim}_{F}M<\infty$ for $F$ a field and $M$ an $F$ vector space. Let $T$ be a linear transformation on $M$. Show that $M$ is cyclic (as an $F[x]$ module) if and only if $m(x)$ ...
0
votes
0answers
34 views

Tensor products, direct sums and products of modules

We get the following problem in our algebra class. Let $ (F_i)_{i \in I} $ be a family of left $R$-modules. a) Define an isomorphism $ \sigma: \oplus_{i \in I} (E \otimes_R F_i) \to E \otimes_R ...
1
vote
0answers
37 views

How to prove A is a flat R-module [duplicate]

Let A be a right R-module. Suppose for every left ideal J of R, the homomorphism $f:A\otimes J\to A$ defined by $f(x\otimes y)=xy$ is injective, then A is a flat R-module.(the identity 1 is in R) I ...
1
vote
2answers
76 views

When can we say elements of tensor product are equal to $0$?

I am learning about tensor products of modules, but there is a question which makes me very confused about it! If $E$ is a right $R$-module and $F$ is a left $R$-module, then suppose we have a ...
2
votes
1answer
32 views

Finding submodules of a module over a ring of matrices

I'm studying for an algebra mid-term and was given this as one of my practice problems. Let $E$ be an n-dimensional vector space over field k. And it has a basis $\{\beta_1, ..., \beta_n\}$. Let ...
2
votes
1answer
32 views

Left adjoint to forgetful from modules to abelian groups

What is the left adjoint to the forgetful functor $U : R-\mathsf{Mod} \to \mathsf{Ab}$? Note here that $R$ is a general ring, not necessarily commutative. I've seen that they define it as $F A = R ...
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vote
0answers
32 views

Minimal number of generators for a f.g module over $\mathbb{Z}[x]$

Let $R = \mathbb{Z}[x]$, $M$ be a finitely generated torsion free $R$-module with rank $n$, and let $\mu ( M)$ denote the minimal number of generators of $M$. For each prime ideal $P \subset ...
0
votes
1answer
52 views

Homomorphism between free modules [closed]

Assume $n>m$. Is it possible to have an injective $R$-module homomorphism from $R^n$ to $R^m$? I feel this is possible, the homomorphism is determined by where we send each basis. So we just ...
0
votes
0answers
59 views

Showing that a submodule of $\mathbb Z$ is free

$\textbf{Definition:}$ A free module is a module that has a basis. If I wanted to show that a $\mathbb{Z}$-submodule of $\mathbb Z$ is free, what would be a way of doing so? Since a ...
1
vote
1answer
53 views

Are those two ways to relate Extensions to Ext equivalent?

Given an extension $\xi$ of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ by taking the long exact sequence $$\ldots\to \operatorname{Hom}(A,X) \to ...
4
votes
1answer
67 views

Equivalent definitions of a root system.

For studying root systems many authors start from a vector space $V$ over $\mathbb{R}$ with a positive definite scalar product $(\cdot,\cdot)$, in which a reflection $\sigma_\alpha$ is a linear ...
0
votes
1answer
37 views

For $R$ a PID, are there any ways to prove that a finitely generated $R$-module $M$ is free iff it is torsion free without the Structure Theorem?

The proof I have in a book for the invariant factor decomposition of an $R$-module over a PID $R$ makes the implicit assumption that for any $R$ PID, there is a free $R$-module $F$. I know that ...