For questions about modules over rings, concerning either their properties in general or regarding specific cases.
1
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0answers
16 views
Identity involving complex sigma function
When trying around with the DivisorSigma function of Mathematica, I found this Identity:
$\#\{a\mid\exists b\in\mathbb{Z}[i]: ab=n\}=\underbrace{\#\{a\mid\exists ...
4
votes
0answers
29 views
a flatness criterion
I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
1
vote
2answers
43 views
Finding invariant factors of finitely generated Abelian group
There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe).
Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...
1
vote
1answer
27 views
Some questions on invariant factor decomposition of modules
Let $M$ be the $\mathbb{Z}$-module $F/N$ where $F=\mathbb{Z}^2$ and $N=\langle(6,4),(4,8),(4,0)\rangle\le\mathbb{Z}^2$. Consider the homomorphism $\varphi:\mathbb{Z}^3\to\mathbb{Z}^2$ such that the ...
1
vote
1answer
38 views
Some questions on Proof of Structure Theorem
I understand the general idea of the proof of Structure Theorem for finitely generated modules over a principal ideal domain but I found it quite difficult to follow some lines of reasoning in the ...
2
votes
1answer
48 views
$\mathrm{Hom}(R/I, R/J\otimes M)\cong ?$
Let $R$ be a Noetherian commutative ring, $I,J$ two ideal of $R$ and $M$ an $R-$module. Does anyone see the isomorphism $\mathrm{Hom}(R/I, R/J\otimes M)\cong \ldots$?
Thanks.
2
votes
0answers
38 views
Structure Theorem for finitely generated modules over PIDs [duplicate]
Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either
$$\begin{pmatrix}
0&-1&0&0\\
...
4
votes
3answers
63 views
example of a flat but not faithfully flat ring extension
I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
7
votes
1answer
41 views
Specific projective dimension of a module over bound quiver
Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver
$$\require{AMScd}
\begin{CD}
1 @>>> 2\\
@V{}VV @V{}VV \\
3 @>>> 4 @>>> 5
...
4
votes
1answer
58 views
How to compute Nakayama functor explicitly?
I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
0
votes
0answers
18 views
Minimal number of generators of $I/IJ(R)$
Let $R$ be a ring and let $I\lhd R$ be a two-sided ideal. The minimal number of generators of $I$ as an $R$-module is clearly at least as large as the minimal number of generators of $I/IJ(R)$ where ...
6
votes
1answer
70 views
is the dual of a finitely generated module finitely generated?
I recently thought of this and have no idea whether over a general commutative ring the dual of a finitely generated module is finitely generated. This must be known.
2
votes
2answers
22 views
length of sum of two submodule
Let $M$ be a $R$-module with finite length and $K$ and $N$ be a submodule of $M$. Prove that
$l(K+N)+l(K\cap N)=l(K)+l(N)$.
My proof: First, by assuming that $K\cap N=\{0\}$, we can conclude that ...
3
votes
1answer
40 views
Finite Projective Dimension implies non vanishing Ext
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$?
Can't we write the free module as a direct ...
1
vote
1answer
19 views
What is the relation between graded modules and finitely generated modules
The reason I ask this question is I found two different statements about Hilbert's syzygy theorem from Jacobson's Basic Algebras 2nd and Wikipedia. Please have a look at the following pictures. The ...
1
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0answers
36 views
$\underset{\longrightarrow}{lim}Hom(R^{n},M_{\lambda}) $ is isomorphic to $Hom(R^{n}, \underset{\longrightarrow}{lim}M_{\lambda})$
Let $R$ be a ring, $\Lambda$ a category of $R$-modules $M_{\lambda}$. Prove the colimit
$$
\underset{\longrightarrow}{\lim}\operatorname{Hom}(R^n, M_\lambda) \cong \operatorname{Hom}(R^n, ...
6
votes
4answers
62 views
$20^{15} + 16^{18}$ is divided by 17
What is the reminder, when $20^{15} + 16^{18}$ is divided by 17.
I'm asking the similar question because I have little confusions in MOD.
If you use mod then please elaborate that for beginner.
...
5
votes
2answers
97 views
When is an element of a free module over a principal ideal domain contained in a basis?
I'm trying to show the following:
Let $R$ be a principal ideal domain and let $M$ be free $R$-module of rank $n$. Let $Y=\{y_1,\ldots,y_n\}$ be a basis of $M$ and $x\in M$ with ...
4
votes
3answers
65 views
Homomorphisms between $ \mathbb{Z} $ modules.
Calculate $\newcommand\Hom{\operatorname{Hom}}\Hom(\mathbb Z \oplus \mathbb Z_{p^\infty},\mathbb Z \oplus \mathbb Z_{p^\infty})$. Where $ \mathbb Z_{p^\infty}= ...
4
votes
0answers
54 views
Artinian rings are perfect
Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
8
votes
2answers
55 views
Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?
Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
3
votes
1answer
26 views
length of composition series
Let $f:M\rightarrow N$ be an injective $R$-module homomorphism. Show that $l(M)\leq l(N)$ where $l(M)$ denote the number of nonzero submodules in a composition series of $M$.
My solution:
Since $f$ ...
5
votes
0answers
90 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
2
votes
1answer
20 views
Module isomorphism, simple modules, and quotients
I'm reading R.S. Pierce's Associative Algebras. While proving a preliminary lemma to Nakayema's Lemma, the following is mentioned:
Let $M$, $N$, be two $A$-modules where $N$ is a submodule of $M$ ...
3
votes
1answer
41 views
Prove $(2, x)$ is not a free $R$-module.
Let $R = \mathbb Z[x]$ and let $M = (2, x)$ be the ideal generated by $2$ and $x$, considered as an $R$-module. Show that $\{2, x\}$ is not a basis for $M$. Show that any 2 elements of $M$ are ...
3
votes
2answers
51 views
$\mathbb{C}[G]$-module homomorphism on finite dimensional modules and finite groups
Nice to meet you folks! I'm currently a grad student reviewing some representation theory of finite groups for prelims next year, and I'm stuck proving a simple statement. Translating the question ...
2
votes
1answer
28 views
Definition of “invariant in a module”
What does it mean if someone say that the class of an ideal $I$ in a ring $R$ is an invariant of a module $M$?
3
votes
1answer
30 views
Sequences of composition factors in composition series
Suppose
$\bullet$ $M$ is a left $R$-module which is both artinian and noetherian,
$\bullet$ $C_1,\ldots,C_k$ is a list of the compositions factors of a composition series of $M$ listed up to ...
1
vote
1answer
42 views
Extension of homorphisms on a divisible R-module
Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
3
votes
1answer
58 views
Does $\mathbb{Z}^{\omega}$ have a weird submodule with a nice basis?
Surprisingly, I think this will be relevant for my algebraic topology work.
Give the module $\mathbb{Z}^{\omega}$ the product topology, and let $M$ be one of its closed submodules.
Let $\mathbf{0}$ ...
0
votes
0answers
57 views
What does “singly generated” mean?
Like the title says, what's the meaning of "singly generated", like in "singly generated Hilbert module". I never came across such terminology and I can't find a definition.
2
votes
1answer
36 views
Annihilator of an element of a left module
How to show that the annihilator of an element of a left module is a left ideal but not necessarily a two-sided ideal. When does this become an ideal?
4
votes
0answers
67 views
Comparison of positive elements and Hilbert C*-modules
I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras.
Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
0
votes
1answer
26 views
The abelian group G is n-divisible if (n,|G|)=1
Consider the abelian group G. If n be a integer number such that (n,|G|)=1 ,Then G is n-divisible.why?
1
vote
0answers
62 views
A counterexample for $\operatorname{Ass}(M_1+M_2)$ [duplicate]
$\newcommand{\Ass}{\operatorname{Ass}}$
Let $A$ be a Noetherian ring and let $M$ be an $A$-module. Suppose $M=M_{1}+M_{2}$, then we have $\Ass(M)\supset \Ass(M_{1})\cup \Ass(M_{2})$. What is an ...
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0answers
42 views
$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact, $M''$ flat. Why is $M$ flat $\Leftrightarrow M'$ flat?
Let $A$ be a commutative ring with identity, let
\begin{align}
0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0
\end{align}
be an exact sequence of $A$-modules, let $M''$ be flat.
I want ...
2
votes
1answer
59 views
Question about minimal projective presentations of a module.
I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 .
On page 108, line 11-14, there is a claim:
If $P_0^{t}\to P_1^{t} \to TrM \to 0$ is not a minimal ...
3
votes
1answer
37 views
Does the projectively stable category have projective modules?
I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1 . On page 109, the projectively stable category is defined by $$ \underline{mod} A = mod A/\mathcal{P}. $$ ...
1
vote
1answer
34 views
Image of the projection map onto an irreducible module
Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
1
vote
1answer
47 views
What's stronger: projective or locally free? flat or locally free?
maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
8
votes
2answers
125 views
What about a module of rank $\frac{1}{2}$?
Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
3
votes
1answer
43 views
Difficulty Understanding Primary Modules
I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
2
votes
0answers
64 views
Help understanding a comment about isomorphisms of two direct sums.
I've recently noticed a few different posts asking about this problem Isomorphism of two direct sums, all using the CRT. I think I understand it this way. What caught my eye is in the above link a ...
2
votes
1answer
80 views
Modules over Principal Ideal Domains
Let $a$ and $b$ be two nonzero elements of a PID $R$. Prove that direct sum $R/(a)\oplus R/(b)$ is isomorphic to the direct sum $R/(u)\oplus R/(v)$, where $u=\gcd(a,b)$ and ...
0
votes
1answer
14 views
Submodules and $p$-adic numbers
I am a little bit confused about the terminology of simple $\mathbb{Q}_p[G]$ module.
E.j.: If one take an $\mathbb{Z}_p[G]$ module $M$, then $pM$ is a submodule, so one can just look for ...
1
vote
1answer
38 views
Disprove injectivity
On searching for some example of divisible module but not injective, I come across one in T.Y.Lam - Lectures on Modules and Rings.
He considers the $\mathbb{Z}[x]-$module $M = ...
0
votes
1answer
31 views
$\operatorname{Res}(V+W)=\operatorname{Res}(V)+\operatorname{Res}(W)$?
if there are two $R[G]$ Modules $V,W$ and $R$ some ring, $S$ subgroup of $G$. Is the formula $$\operatorname{Res}_S (V \oplus W) = \operatorname{Res}_S (V) \oplus \operatorname{Res}_S (W) $$
true? I ...
1
vote
1answer
85 views
Artinian ring and faithful module of finite length
Let $A$ be a ring. How can I prove that:
$A$ is an Artinian ring $\Leftrightarrow \exists$ a faithful $A$-module which is of finite length.
I know that if a ring has a faithful $A$-module which ...
0
votes
0answers
22 views
Computation of $\hom_\mathbb{Z}(\mathbb{Z}_{p^\infty},\mathbb{Q})$
I'm trying to compute $\hom_\mathbb{Z}(\mathbb{Z}_{p^\infty},\mathbb{Q})$. I believe it is zero, simply because $\mathbb{Z}_{p^\infty}$ is torsion (it is a $p$-group) and $\mathbb{Q}$ is torsion-free.
...
2
votes
1answer
24 views
Submodules of an $R$-module where $R$ is the set of $n\times n$ upper triangular matrices
Let $R$ be the ring of $n\times n$ upper triangular matrices with coefficients in a field $K$. Let $V$ be the $R$-module consisting of all $1\times n$ matrices with coefficients in $K$. Define $$V_r = ...
