Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
6,026 questions
0
votes
1answer
28 views
Prove $(F/R) \otimes_R (F/R)=0$ for $R$ an integral domain and $F$ field of fraction
Suppose $R$ is an integral domain and $F=Frac(R)$ as an $R$-module. I want to prove that
$$(F/R) \otimes_R (F/R)=0$$
My attempt:
I first considered the exact sequence
$$R \rightarrow F \rightarrow ...
-1
votes
1answer
27 views
In which I forget what a module is.
In section 10.4 of D+F, in Theorem 8, there is the following use of the universal property of the free module.
We have that $R$ is a subring of $S$, and $N$ is a left R-module, and L is a left S-...
1
vote
0answers
29 views
Proof-verification: Show that there exists an exact sequence
Let $R$ be a ring and let
$$
M_1 \stackrel{f_1}{\hookrightarrow} M_2 \\
\alpha_1\downarrow \hspace{1cm} \downarrow \alpha_2 \\
N_1 \stackrel{f_2}{\hookrightarrow} N_2
$$
be a commutative diagram of $...
0
votes
0answers
34 views
Is $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ free?
I'm trying to solve a question which asks me to determine whether the quotient $\mathbb{Z}$-module $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is free. I'm then supposed to find some ...
1
vote
0answers
18 views
Finding the submodules of the $\mathbb{R}[x]$-module defined by $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & -1 \end{bmatrix} $
I'm trying to solve a question which asks me to consider the matrix $A$ with
$$A=\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 &0 & -1
\end{bmatrix}
$$
...
0
votes
1answer
25 views
Counterexample: surjective module homomorphism in an exact sequence
I need to give an example to show that the exactness of $0 \to M_1 \to M_2 \to M_3 \to 0$ need not imply that the map $\operatorname{Hom}_R(N,M_2) \to \operatorname{Hom}_R(N,M_3)$ is surjective, where ...
-1
votes
1answer
14 views
Show that the module of fractions of a projective module is projective. [on hold]
Let $R$ be a commutative ring (with identity). Let $S$ be a multiplicative subset of $R$, and $P$ a projective $R$-module. Prove that $S^{-1}P$, the module of fractions of $P$ with respect to $S$, is ...
6
votes
1answer
26 views
Cokernel in Categories of Module (Exercise 4.4 in Blyth's book)
i have been working on "Module Theory:An Approach to Linear Algebra" by T. S. Blyth and i am stuck on exercise 4.4 which is
"Let $f: M \to N $ be an $R$-morphism. By a cokernel of $f$ we mean a pair $...
0
votes
0answers
20 views
Big picture of module theory [duplicate]
This might a question that is too vague or broad, but I'm trying to understand the big picture of module theory. In my algebra class, we have covered things like free modules, injective modules, and ...
1
vote
0answers
17 views
Skew-symmetric implies alternating for $2$ a zero divisor in $R$?
In Keith Conrad's notes: https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf
Theorem 2.10 reads:
Let $k\geq 2$. If $2\in R^\times$, then a multilinear function $f:M^k \to N$ which is
...
0
votes
2answers
29 views
Find the values of $a\in \mathbb{Z}[i]$ such that $(2,1)$ and $(2+i,a)$ form a basis of $\mathbb{Z}[i]^2$.
I'm trying to solve an exercise which asks me to determine for what values of $a\in \mathbb{Z}[i]$ $(2,1)$ and $(2+i,a)$ form a basis of $\mathbb{Z}[i]^2$ (where we're considering $\mathbb{Z}[i]^2$ as ...
3
votes
1answer
52 views
Showing that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are not isomorphic as $\mathbb{R}[x]$ modules.
I'm trying to solve an exercise which asks me to prove that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are isomorphic as rings, but not as $\mathbb{R}[x]$...
8
votes
0answers
113 views
+100
Corollary of the Malgrange Preparation Theorem
Let $f:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ be a smooth function, such that
$$f(0,0)=0,\ \frac{\partial f}{\partial t} (0,0) = 0,\ldots, \frac{\partial^{k-1} f}{\partial t^{k-1}} (0,0) = 0,\ \...
0
votes
0answers
17 views
Representing certain linear maps on a direct sum decomposition of a free module of finite rank
Suppose that $R^n \cong M \oplus N$ where $R$ is a commutative ring and $M$ and $N$ are $R$-modules.
Consider a map of the form $f \oplus 0_N: M \oplus N \to M \oplus N$ where $f: M \to M$ is a ...
1
vote
0answers
40 views
Necessary and sufficient condition of injectivity for a module based on exact sequences
Show that Q is injective if and only if, whenever $$0\to A\stackrel{f}{\to} B\stackrel{g}{\to} C\to 0$$ is exact, then $$0\to \mathrm{Hom}_R(C,Q)\stackrel{g^*}{\to} \mathrm{Hom}_R(B,Q)\stackrel{f^*}{\...
0
votes
0answers
26 views
Show a sequence of R-modules and R-homomorphisms is exact. [duplicate]
Suppose given a exact sequence, $$0 \to A\xrightarrow{\enspace f\enspace} B\xrightarrow{\enspace g\enspace} C\to 0$$
I want to show the sequence,$\DeclareMathOperator{\Hom}{Hom}$
$$0→\Hom_R(N,A) \...
2
votes
1answer
25 views
Module homomorphism is well defined
Let M be R-module, and A,N submodules such that $A \subset N$
Let $f:M/A \to M/N$ be defined by $f(mA)=mN$ for $m\in M$.
I'm having trouble seeing why exactly this map is well-defined (It should be). ...
6
votes
0answers
59 views
Maps between short exact sequences
Suppose I have a short exact sequence of modules $0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0$. Let $A' \subseteq A$ and $C'\subseteq C$ be submodules and suppose I have a short exact ...
1
vote
1answer
38 views
If $ \operatorname{Ass}(M)= \operatorname{Assh}(M)$, then $M$ is a Cohen-Macaulay $R$-module? [duplicate]
Let $R$ be a local commutative Noetherian ring and $M$ a finitely generated $R$-module. We denote by $ \operatorname{Assh}(M)=\{ \mathfrak{p}\in \operatorname{Ass}(M) \mid \dim R/\mathfrak{p}=\dim M\}...
2
votes
1answer
20 views
Tensor product of a free module commutes with arbitrary intersection
I was wondering if the following holds true:
Suppose that we have two modules $M$ and $N$ over a commutative ring $R$. Suppose that $M$ is a finitely generated free $R$ module. Suppose $(N_i)_{i\in ...
1
vote
0answers
23 views
Set of all R-homomorphism is finitely generated
I have the following proposition:
Proposition: If $R$ is a principal ideal domain (PID) and $M$, $N$ two finitely generated (fg) $R-$modules, then $\text{Hom}_R(M,N)$ is finitely generated.
My idea: ...
0
votes
0answers
29 views
Let $M_i, i \in I$, be R-modules. Show that the (external) direct product $\prod_I M_i$ satisfies the following universal property
Let $M_i, i \in I$, be R-modules. Show that the (external) direct product $\prod_I M_i$ satisfies the following universal property relative to the R-homomorphisms $\pi_j:\prod_I{M_i} \to Mj$ defined ...
2
votes
0answers
19 views
Question about a group algebra being not Artinian
Let $F=GF(2)$ and $G=\langle a_1,a_2,\ldots|a_i^3=1,[a_i,a_j]=1\text{ for }i,j\in\mathbb{N}\rangle$, that is the direct product of infinitely many cyclic groups of order $3$. I want to show that the ...
-1
votes
0answers
21 views
A difficulty in understanding a part of the definition of tensor product.
when I searched about the definition of Tensor product of modules on Wikipedia I found the following sentence:
"In mathematics, the tensor product of modules is a construction that allows arguments ...
0
votes
0answers
45 views
Why is $k[x_1] \to A$ not finite and $\phi(y)=x_1+x_2$ finite?
I was reading https://www.math.columbia.edu/~dejong/courses/deJongNotes.pdf. (Observe you don't have to look at the link as the question is self contained)
See Example 1 in the beginning. There I ...
0
votes
2answers
26 views
A difficulty in understanding the proof of distributivity of tensor products over direct sums for modules.
Here is the proof:
But I do not understand the following:
1-why the function needed to be bilinear to use the universal property?
2- what is he doing starting from the paragraph that starts with ...
0
votes
1answer
52 views
A difficulty in understanding the universal property of modules.
The property is given below ( from Dummit & Foote) but I have a difficulty in understanding why it is universal property and what is its importance or when usually we use it?and why the function ...
1
vote
1answer
31 views
Question on modules of finite length
Let $K$ be a field and let $f$ be a nonzero polynomial on $K$. Then $K[X]/\langle f\rangle$ is a both a $K$-module and a $K[X]$-module. It can be shown that $\ell_{K[X]}(K[X]/\langle f\rangle)\leq \...
0
votes
0answers
28 views
How to find all submodules of a given $\Bbb R[x]$-module?
In class my professor gave an example where $M$ is the $\mathbb R[x]$-module given by $\mathbb{R^{3}}$ with $x$-action given by
$$A =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & ...
-2
votes
1answer
35 views
Proper $\Bbb Z$-submodules of $\Bbb Q$ are finitely generated or not? [closed]
Let $M$ be a proper $\Bbb Z$-submodule of $\Bbb Q.$ Can we say that $M$ is finitely generated?
I know that $\Bbb Q$ is not finitely generated as a $\Bbb Z$-module.
Please help me in this regard. ...
0
votes
0answers
12 views
Do we have rings for which $(I:a)=(0:a)$ or for which $(I:a)\cong(0:a)?$
I have been reading a book on Macoy rings and I saw these definitions for the annihilator sets: let $R$ be a ring and $I$ an ideal of $R$.
$$(I:a)=\{r\in R: ra\in I\}~~\text{and}~~(0:a)=\{r\in R: ra=...
0
votes
0answers
20 views
Show that V is a submodule of the A-module from a given quiver
Let $Q = 1\leftarrow 2 \leftarrow 3$ Where the arrows are labeled $a$ and $b$, and $A=kQ$
a) Let V be the subspace of A on basis {$e_2$, α}. Show that V is a submodule of the A-module $_AA$ equal
to ...
0
votes
0answers
11 views
Showing A submodules from a product of k-algebras
Let $A = A_1 × A_2$ be the direct product of two k-algebras and M an A-module.
Set $M_1 := \{(1_{A1} , 0) m : m ∈ M\}$ and $M_2 := \{(0, 1_{A2} ) m : m ∈ M\}.$
Show that $M_1$ and $M_2$ are A-...
0
votes
0answers
23 views
Showing $M_1$ and $M_2$ are submodules of $M$, where $M$ is a module of a direct product algebra.
Let $A = A_1 \times A_2$, the product of two $k$-algebras.
Suppose $M$ is some A-module.
Define $M_1 \triangleq \{(A_1,0)m : m \in M\},\ M_2 \triangleq \{(0,A_2)m : m \in M \}$.
Show that $M_1$ ...
3
votes
1answer
31 views
The “span” of a $\mathbb{Z}$-module.
Consider the $\mathbb{Z}$-module $M=\mathbb{Z}^2$,
$$
M_1=\{(a_1,a_2):2a_1+3a_2=0\} \text{ and } M_2=\{(a_1,a_2):a_1+a_2=0\}$$
Prove that $M$ is the internal direct sum of $M_1$ and $M_2$.
To prove ...
3
votes
1answer
35 views
Turning the Ring of Polynomials, $\mathbb{F}[\lambda]$, into a Module.
Modules associated to a linear operator. Suppose that $\mathbb{F}$ is a field and $V$ a vector
space over $\mathbb{F}$( i.e. an $\mathbb{F}$-module). Let $T:V\rightarrow V$ be a linear operator on $V$ ...
1
vote
0answers
19 views
Any vector shorter than $\lambda_1^*$ must belong to a sublattice.
Let $R = \mathbb{Z}[x] / \langle x^n -1 \rangle$ and $f, g, F,$ and $G$ be polynomials in $R$.
Let $\Lambda_h = \{ (f, g)u + (F, G)v : u, v \in R \}$ be the $R$-module (lattice) generated by linear ...
2
votes
1answer
44 views
$\mathbb{Z}$-module $\prod\limits_{\text{p prime} } \mathbb{Z}/p\mathbb{Z}$
I am trying to establish if the $\mathbb{Z}$-module $\displaystyle\prod_{\text{p prime} } \mathbb{Z}/p\mathbb{Z}$ is torsion-free. So I think that the elements of $\displaystyle\prod_{\text{p prime} }...
0
votes
1answer
37 views
$M$ and $N$ flat, then $M\otimes N$ flat
I want to show that if $M$ and $N$ are flat $R$-modules, then $M\otimes_R N$ is flat.
By flat we mean that if $0\to A\to B$ is exact, then $0\to A\otimes_RM\to B\otimes_RM$ is exact.
I am assuming ...
1
vote
1answer
28 views
When do $M_n, GL_n, SL_n$ commute with direct products?
We know, for example, that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$. To what extend does this hold in general? That is, if we're given, say, some commutative ...
0
votes
0answers
38 views
Axes of a free module over a PID (2)
Let $R^n$ be a finitely generated free module of rank $n > 0$ over a principal ideal domain.
I am trying to prove that for every non-zero element $a$ of $R^n$ there is a basis such that $a$ ...
1
vote
1answer
28 views
Simple question about flat modules
Everything I can find about this is stated in category theory language that I do not understand. If I have an exact sequence
$... \rightarrow A \xrightarrow[\text{}]{\text{f}} B \xrightarrow[\text{}]{...
1
vote
0answers
13 views
Let $r\in R$ and let $B$ be any $R$-submodule of a right $R$-module $A$. Then $A/(Ar+B)\cong A/Ar$.
Let $r\in R$ and let $B$ be any $R$-submodule of a right $R$-module $A$. Then $A/(Ar+B)\cong A/Ar$.
In the proof, I have defined the map $f:A\to A/Ar$ by $f(a)=a+Ar$ for all $a\in A$. $f$ is well ...
0
votes
1answer
85 views
finitely generated projective module and Nakayama's lemma
Let $R$ be a local ring with maximal ideal $I$. $M$ is a finitely generated module over $R$ generated by $a_1, \ldots, a_n$ and the generators are chosen such that their quotients in $M/IM$ form a ...
1
vote
1answer
58 views
Defining the cycle functor $Z_* : \mathbf{Ch_R} → \mathbf{GrMod_R}$
In example (viii) of section 1.3.2 of Category Theory in Context, the $n$-cycle functor is defined on objects as
\begin{align}
Z_n: \mathbf{Ch_R}&\to\mathbf{Mod_R} \\
C_\bullet&\mapsto \ker(...
3
votes
1answer
69 views
$\mathbb{Z}$-Module exercise
I am trying to solve the following exercise on basic module theory and I am stuck. Any help would be more than welcome!
So let $M\subseteq \mathbb{Z}^3$ the solutions to the following problem:
$-3x+...
0
votes
0answers
13 views
Pathological examples of finitely generated modules
Let $R$ be a commutative (noetherian) ring with identity. Let $M$ be a finitely generated $R$-module. Let $n\in\mathbb{N}$. Does $$M=\bigoplus_{k=1}^n M_k,$$ where $M_k$ is an either an ideal or ...
3
votes
0answers
45 views
Are $\mathbb{Z}$ and $\mathbb{Z}_n$ the only rings (with identity) whose modules are equivalent to abelian groups?
Let $R$ be a ring with identity. Let $M$ and $N$ be $R$-modules. Let $f$ be an (arbitrary) group homomorphism from $M$ to $N$. Under what conditions on $R$,$M$, and $N$ is $f$ also a $R$-module ...
4
votes
1answer
60 views
Show that $M[x] \cong A[x] \otimes_{A} M.$
I'm trying to solve the problems in the book of Atiyah and MacDonald. I want to verify my solution to the problem 2.6.
This is the exercise's statement:
2.6. For any $A$-module $M$, let $M[x]$ ...
2
votes
1answer
79 views
Axes of a free module over a PID
Let $R^2 = R \times R$ be a free module of rank $2$ over a principal ideal domain.
I am trying to prove that for every non-zero element $a$ of $R^2$ there is a basis such that $a$ belongs to one of ...
