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

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4
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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
33 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
19 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
67 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
47 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
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2answers
27 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
29 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
74 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 + ...
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0answers
28 views

Suppose $R$ is commutative integral domain. If $M$ is a vector space over $Frac(R)$, then it is divisible and torsion free as an $R$-module.

My question concerns part (1) of the exercise below (Exercise 3.15(p. 54) in "Rings and Categories of Modules" F.W. Anderson and K.R Fuller.) I have included the text, and my attempted solution ...
2
votes
1answer
31 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
12 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
40 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
33 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
64 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
31 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
29 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
51 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
57 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
52 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
64 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
34 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 ...
0
votes
1answer
39 views

Finding an example of an extension of length 2

I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an ...
2
votes
1answer
56 views

How to prove the following sequence is exact?

Let R be a ring and $F',F,F'',G',G,G''$ left R-modules. Assume we are given R-module homomorphism $i:F'\to F,p:G'\to G,p':G\to G''$ and $a:F'\to G',b:F\to G,c:F''\to G''$ such that the following ...
1
vote
3answers
127 views

Bijection between Extensions and Ext (Weibel Theorem 3.4.3)

I was wondering about one step in the proof of surjectivity of $\Theta$ constructed for Theorem 3.4.3 in Weibel's "An introduction to homological Algebra". For an extension $\xi:0\to B\to X\to A\to0$ ...
3
votes
1answer
48 views

Proof of Tensor Products from Atiyah and Macdonald's Commutative Algebra

I apologize in advance if this question is stupid, but I am self studying and I am having trouble deciphering the following notation in Atiyah and Macdonald's "Introduction to Commutative Algebra." ...
1
vote
1answer
25 views

$M$ is the (possibly infinite) direct sum of its $p$-primary components as $p$ runs over all primes of $R$.

Let $R$ be PID, let $M$ be a torsion $R$-module and $p$ be prime in $R$. Prove that $M$ is (possibly infinite) direct sum of its $p$-primary components as $p$ runs over all primes of $R$. Take $m \in ...
0
votes
1answer
36 views

Question about injectivity of tensor products

Let $A$ be a commutative ring, $M$ a module over it and $k_1,k_2$ fields such that we have the following maps $A \to k_1\to k_2$. Construct the natural map: $$f: M\otimes_A k_1 \to M\otimes_A k_2.$$ ...
2
votes
0answers
32 views

kernel and image submodules proof

Let $f: V → W$ be a homomorphism of $G$ modules. Show $kerf $ is a $G$ submodule of $V$, and $imf$ is $G$ submodule of $W$ . proof: Since $v$ is linear , then we only need to check for closure ...
1
vote
2answers
37 views

$M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences [duplicate]

Let $0 \to M_1 \to M \to M_2 \to 0$ be an exact sequence of $A$-modules. i) Prove: If $M_1$ and $M_2$ are finitely generated, then $M$ is too. ii) Let $M$ and $N$ be sub-modules of an $A$-module ...
2
votes
1answer
35 views

Tensor left adjoint as a bi-functor?

Working in the category of modules over a fixed ring $A$ $\operatorname{Mod}_A$, we have the adjunction: $$F(X) = X\otimes_A M \dashv G(X) = \operatorname{Hom}_A(M,X)$$ for a fixed module $M$. Is ...
1
vote
1answer
27 views

Reference request for modules over polynomial rings

We didn't cover these in class but have homework about them, but what do I call "$F[x]$-modules"? (how do I say it out loud) Modules over polynomial rings? What do I google to find lecture notes about ...
1
vote
1answer
28 views

$M_m = 0$ for all maximal ideals $m \in R$ implies $M=0$?

Suppose $M$ is an $R$-module, and suppose that the localization of $M$ at $m$, $M_m = 0$ for all maximal ideals $m \in R$. Show that $M=0.$
0
votes
1answer
30 views

Categories of left $R$-module in which every $R$-modules is injective

I have some questions I would like to answer. Both of them are related to injective modules. First, I want to clarify that I am not interested in the following categories as options to answer my ...
6
votes
1answer
112 views

Does $f\otimes \operatorname{Id} = \operatorname{Id}$ imply $f= \operatorname{Id}$?

Let $R$ be a commutative ring, and $X$ an $R$-module. If an $R$-endomorphism of $X$ satisfies $f\otimes \operatorname{Id}_X = \operatorname{Id}_{X\otimes X}$, is it true that ...
3
votes
1answer
47 views

On characterizing modules that don't annihilate any module under tensor product.

Let $R$ be a commutative ring and $M$ an $R$-module. Then under what condition can we deduce that for any nonzero $R$-module $N$, $M\otimes_RN\neq0$?
0
votes
3answers
44 views

Free $R$-module question

Let $R$ be a ring with identity element. Let $M$ be a finitely generated $R$-module. Show that there is a free $R$-module $F$ and a submodule $K\subseteq F$ such that $M\cong F/K$ as $R$-modules. ...
6
votes
2answers
249 views

Is it always possible to find one non-trivial homomorphism between modules?

I believe this question is a elementary one, and it may have a very simple answer, of which I'm not aware yet. Given two non-trivial modules over the same non-trivial ring (or two groups, or two ...
0
votes
3answers
39 views

Binomial coefficients of $p^a-1$ mod p

I need to show that $$\binom{p^a-1}{k}\equiv (-1)^k\mod p$$ where $p$ is a prime, $a$ is a positive integer, and $k<p^a-1$. I think I have done most of the proof, but I'm stuck at the very end. ...
0
votes
0answers
33 views

Show that $D^n$ is simple and unique [duplicate]

Suppose $A=M_n(D)$, i.e. n by n matrices over a division algebra $D$. We want to prove that: $D^n$ (viewed as row vectors) is a simple right $A$-module and any other simple $A$-module is ...
1
vote
1answer
101 views

Doubt about a kernel

I was reading the proof of Lemma 1.25 in this thesis and I thought I understood it, but I think I don't. The thing that I don't see clearly is in page 26 where he is showing that $\textrm{ker}\ ...
0
votes
0answers
37 views

Showing that an element generates the kernel

$I $ is a monomial ideal generated by $\left < m_1, \dots, m_n\right >$ and suppose we also have an $R$-module homomorphism $\phi: \oplus_{j = 1}^n Re_j \to I$ defined by $$\phi(e_i) = m_i.$$ ...
0
votes
0answers
47 views

The $R/I$-module $F/IF$.

Let $F = \oplus_{i = 1}^r Re_i$ be a free $R$-module and $I\subset R$ be an ideal and $IF = \{ if :i \in I, f \in F \}$ is a submodule of $F$. Then (1) $F/IF$ is an $R/I$-module and (2) $\{e_i ...
1
vote
1answer
20 views

Proof of Quaternion Algebras being Division Algebra or $M_2(F)$

I have a question regarding the proof of quaternion algebras (denoted $A$) being either a division algebra or isomorphic to $M_2(F)$. I can understand that $A$ is central simple. And since $A$ is ...
1
vote
1answer
63 views

Show that a sequence is a free resolution

Let $I \subset R = k[x_1,\dots,x_n]$ be an ideal and $f \in R$ such that $I = \left < f \right >$ ($k$ is a field, so R is commutative ring). How do I show that (1) $I$ has a free resolution ...
1
vote
1answer
26 views

Is this a semisimple algebra?

Let $A$ be the set of 2 by 2 matrices of the form $\begin{pmatrix}d_1 &d_2\\d_3 & d_4\end{pmatrix}$, where $d_i$ are in division algebras $D_i$. Update: $D_1=Hom(N_1,N_1)$, ...