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

learn more… | top users | synonyms

0
votes
1answer
12 views

Simple modules over R isomorphic to R/I

Let $R$ be a ring, and let $M$ be a simple $R$-Module, meaning that it only has the trivial submodules {0} and $M$. Show that there's a maximal ideal $I \subset R$ so that $M \cong R/I$. Thanks in ...
0
votes
0answers
19 views

Zorn's Lemma Application for Finding Maximal Submodule?

I came across the following exercise in my algebra textbook. Let $M$ be a left $R$-module ($R$ is ring with unity $1_R$) and let $N\subseteq M$ be a submodule such that $x\in M-N$ for some $x\in ...
0
votes
0answers
19 views

Relation between two definitions of primary modules

Let $A$ be a commutative ring, $M$ be an $A$-module and $N \leq M$. There are two definitions of primary modules: 1) $M/N$ is coprimary (i.e., every zero divisor is nilpotent); 2) $\text{Ann}_A(N)$ ...
0
votes
0answers
12 views

Are there terminologies distinguishing these two ranks?

Definition 1 Let $R$ be a ring with invariant basis number and $M$ be a free $R$-module. Then, the rank of $M$ is the cardinality of an $R$-basis of $M$. . Definition 2 Let $R$ ...
0
votes
0answers
23 views

One-dimensional submodules of $\mathbb{C}^4$; direct sum of submodules.

I'm having some trouble understanding my lecture notes. I need to find a one-dimensional submodule $U$ of $\mathbb{C}^4$. Is $u=1+x+x^2+x^3$ valid? My reasoning: because $ux = x + x^2 + x^3 + 1 = u ...
4
votes
2answers
41 views

Is $m$ a projective $A$-module?

$A$ is a Noetherian local ring and $m$ be its maximal ideal. Then is $m$ a projective $A$-module? I got this problem while solving another problem. Can anyone please help me to figure it out?
2
votes
3answers
85 views

Is there an counterexample for the claim: if $A \bigoplus B\cong A\bigoplus C$ then $B\cong C$

Here A, B and C are R-modules. Is there an counterexample for the claim: if $A \bigoplus B\cong A\bigoplus C$ then $B\cong C$? And what if B and C are finitely generated?
1
vote
0answers
31 views

some exterior power of a lattice, torsion-free? (about a remark of Serre in Local Fields)

I have a question on a remark of Serre in chapter III §2 of his book Local Fields (p. 49). In this section he wants to define the discriminant of a lattice with respect to a bilinear form. The ...
3
votes
1answer
38 views

Prove that the following is a non zero tensor.

I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module. My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map ...
1
vote
1answer
22 views

What is the free module of this one?

Let $R=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ and $A=\{(1,0)\}$. Then, what is the free $R$-module $F(A)$? (Here the module action is mutiplying componentwise. ) I tried spanning $A$ by ...
1
vote
1answer
23 views

Is torsion-free equivalent to free for non-finitely generated modules over a PID?

Maybe this is a trivial question. If $A$ is a PID and $M$ is a finitely generated $A$-module, it's well known that $M$ is torsion-free iff $M$ is free. However, if $M$ is not finitely generated, does ...
1
vote
1answer
23 views

Are the Eisenstein integers the ring of integers of some algebraic number field? Can this be generalised?

The Eisenstein integers are $\mathbb Z[\omega]$ where $\omega$ is the primitive third root of unity. If $K$ is some algebraic number field, can $\mathcal O_K$ be isomorphic to $\mathbb Z[\omega]$? ...
0
votes
0answers
21 views

$ M \in R\text{-Mod} $ is pure-injective if and only if there is $ N \in R\text{-Mod} $ such that $ M $ is a direct summand of $ N^{*} $.

I found this question in the exercises of the first chapter of Stenstrom’s book Rings of Quotients. “$ M \in R\text{-Mod} $ is pure-injective if and only if there is $ N \in R\text{-Mod} $ such that ...
2
votes
1answer
16 views

Set of generators of a $R$-algebra?

In my algebra textbook I came across the following: Let $A$ be a $R$-algebra ($R$ is a commutative ring with unity $1_R$) and suppose $X\subseteq A$ a generator set for $A$. In this context, ...
2
votes
2answers
46 views

When can the rank of a submodule be bigger than the rank of the module itself?

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring. I am looking ...
3
votes
1answer
34 views

$\mathbb{Z}/p^n$ is an injective module over itself. [duplicate]

For any prime power $p^n$, prove that $\mathbb{Z}/p^n$ is an injective module over itself.
1
vote
1answer
13 views

Representation matrix for modules map

Here $S=\mathbb Q[x,y]$, and we define $\oplus Se_i$ to be a $S$-free module with basis $\{e_1,e_2,e_3\}$. Define a map from $\oplus Se_i$ to $S$ by $e_1\to x^2$, $e_2\to xy+y^2$, $e_3\to y^3$. Is the ...
0
votes
1answer
32 views

Basis of a Z-module

I think I might know how to start this problem but I'm not sure how to finish. Here is the statement: Determine a basis for the ℤ-module of integer solutions to the following system of equations: ...
0
votes
1answer
65 views

Short exact sequence of modules over a Noetherian local ring of depth $1$.

I am reading an article in algebraic geometry and am having trouble understanding a particular point that reduces to a problem in commutative algebra. I'm not familiar with the concepts involved so am ...
0
votes
1answer
11 views

quotient of lattices, why of finite length? (about a statement in Local Fields of Serre)

I am studying the book "Local Fields" from Serre. At the beginning of chapter III the setting is follows: $A$ denotes a Dedekind domain, $K$ its field of fractions and $V$ is a finite-dimensional ...
1
vote
1answer
14 views

$_RM_S$ is R-projective and $_SN$ is S-projective then $_RM \otimes_S N$ is R-projective.

I want to prove that if $_RM_S$ is R-projective and $_SN$ is S-projective then $_RM \otimes_S N$ is R-projective. Projectivity of $_RM$ and $_SN \implies M \oplus K \cong R^{(I)}$ and $N \oplus L ...
7
votes
0answers
40 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
1
vote
1answer
41 views

Why $R/AB$ is cyclic?

Why $R/AB$ is cyclic when $A,B$ are ideals of $R$? I know a cyclic module is a module that can be written as $Rm$ and $Rm$ is isomorphic to $R/Ann(m)$. However, I can't see why the module is ...
0
votes
1answer
13 views

Kernel of a $R$-linear map is uniquely determined?

In my algebra textbook there is the following exercise: Let $M$ and $N$ be two left $R$-modules and $f:M\longrightarrow N$ a $R$-linear map. Show the kernel $L$ of $f$ is uniquely determined by the ...
1
vote
1answer
29 views

Smith Normal Form and quotient $\mathbb{Z}^{3}/M \mathbb{Z}^{3}$

I am learning modules and the Smith Normal Form, but I got stuck in the following: I found the Smith Normal Form of $$M = \begin{pmatrix} 21 & 0 & 1 \\ 8& 4 & 1\\ 3& 8 & 1 ...
0
votes
0answers
17 views

Show that infinite direct sum of injective modules is not an injective module [on hold]

I want an example to show that infinite direct sum of injective modules is not injective
2
votes
1answer
38 views

Show that if $R$ is commutative, then $\mathrm{Ann}(n_1) + \mathrm{Ann}(n_2) = R$.

Let $R$ be a commutative ring with $1$, and let $M$ be a cyclic $R$-module with generator $m$ (so that $M=Rm$). Suppose that $M = N_1 \bigoplus N_2$ for some submodules $N_1$ and $N_2$ of $M$. Let ...
1
vote
1answer
35 views

Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
0
votes
1answer
19 views

$P$ is projective . Show that $P$ is a direct summand of a free $R-$ module

$P$ is projective . Show that $P$ is a direct summand of a free $R-$ module I am using the definition of a projective module as $P$ is projective if every exact sequence $M\rightarrow P\rightarrow ...
2
votes
1answer
32 views

Prove that if $M$ is a simple left $R$-module, $Ann(m)$ is a maximal left ideal of $R$ and $M \cong R/Ann(m)$ for all $m \in M\backslash \{0\}$

I need to prove equivalent statements of a simple left $R$-module. However, I'm stuck at the following: The map $\phi_m: R/Ann(m) \rightarrow M$, such that $\phi_m(r+Ann(m)) = rm$ for all $r + ...
0
votes
0answers
40 views

Find all non-trivial submodules of a direct sum of two non-isomorphic simple modules

Let $R$ be a ring with $1$. Let $M_1$ and $M_2$ be two non-isomorphic simple (nonzero) $R$-modules. Find all non-trivial submodules of $M_1 \bigoplus M_2$. Solution: $M_1 \bigoplus M_2 \cong M_1 ...
0
votes
1answer
36 views

Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2, \dots, x_n, \dots$. (Of course, each element of $R$, being a polynomial, will involve only ...
2
votes
2answers
53 views

Direct sum of two non-zero $R$-modules

If $R$ is a commutative integral domain, how can I show that it cannot decompose as a direct sum of two non-zero $R$-modules (when viewed as a module over itself). If $n\geq 1$, is there an example ...
0
votes
1answer
25 views

Quick question about mod equation

So here is the mod function: $$5 ^ {31} \cdot 2 ^{789} - 23^{23}\pmod{10}$$ Is there a way to shorten it, or I must calculate it plain numbers? I have tried the mod powers rule, but except for the ...
1
vote
1answer
58 views

Question on a property of $\mathrm{Ass}(M)$ for modules over noetherian rings

I got stuck reading a proof of the following lemma (Lemma 0.19 in this file): Lemma Suppose that $M$ is a module over a commutative noetherian ring $R$ and let $m\neq 0 \in M$. Let $S$ be a ...
0
votes
0answers
29 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
1
vote
0answers
26 views

Maximal ideals of the ring of matrices

Let K be a field. We consider $K^n$ as a left module of $M_{n, n}(K)$, the ring of matrices of size $n$ over $K$. 1) For any $M_{n, n}(K)$ module homomorphism $ 0 ≠ \phi: M_{n, n}(K) \to K^n$, show ...
1
vote
1answer
25 views

Finitely generated modules and submodules

Let $R$ be a ring, $M$ an $R$-module and $U$ a submodule of $M$. Show that, if $U$ and $M/U$ are finitely generated, then $M$ is finitely generated aswell. I thought to maybe show this by taking ...
2
votes
3answers
44 views

Show that $N = f(M) \bigoplus N'$ if and only if there exists $\alpha: N \rightarrow M$ such that $\alpha(f(m)) = m$ for all $m \in M$.

Let $R$ be a ring with $1$, and let $f: M \rightarrow N$ be an $R$-module homomorphism. Suppose that $f$ is injective. Show that $N$ has a submodule $N'$ such that $N = f(M) \bigoplus N'$ if and only ...
0
votes
0answers
21 views

Module length and connection between $\varphi,\det\varphi$ [closed]

$e_A(\varphi,M) = l_A(\mathrm{coker}\varphi) - l_A(\ker\varphi)$ Let $A$ be domain. I want to prove that $e_{A}(\varphi,A^n) < \infty \iff e_{A}(\det\varphi,A) <\infty$ using the fact that ...
-1
votes
0answers
24 views

direct sum of modules is isomorphic to the direct sum permuting indices? [closed]

the primary for the exercise idea is to use the universal property of the external direct sum of modules.
1
vote
1answer
68 views

Determine a basis for $\mathbb{Z} \oplus \mathbb{Z}$ which determines a basis for the submodule $N$ generated by $(6,9)$

Proof Clearly the rank of $\mathbb{Z} \oplus \mathbb{Z}$ is $2$, so we must have that the rank of $(6,9)$ is $\leq 2$.Let $e_1 = (1,0)$ and $e_2 = (0,1)$ be a basis for $\mathbb{Z} \oplus ...
3
votes
1answer
45 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
1
vote
1answer
14 views

Isomorphism of tensor product involving a principal ideal

This question arose when dealing with a long exact sequence of Tor. Let $R$ be a (not necessarily commutative) ring, $g$ a central element of $R$ and $M$ a right $R$-module. We have an exact sequence ...
0
votes
2answers
41 views

Factorization in modular arithmetic

Is this expansion a legal step? $12^8\mod15 = 2^8 * 2 ^ 8 * 3 ^ 8\mod15$
0
votes
0answers
37 views

Property of free submodules for a module over a PID [duplicate]

This question was asked here and remains without solution. It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...
0
votes
0answers
13 views

Extra Operation required for Smith Normal Form over PID-Theoretical Justification

Why does one need an extra operation for performing smith normal form over a PID? One might suspect and say that it is because of the lack of Euclidean algorithm or just say that we need the ...
1
vote
2answers
52 views

Module structure of base extension via tensor product

Let $A,B$ be commutative rings. Defining a product of $B\otimes_{A}B$ as $(b_1 \otimes b_2)\cdot (b_3 \otimes b_4)=(b_1b_3)\otimes(b_2b_4)$, this becomes a commutative ring. Defining $b\cdot(b_1 ...
1
vote
1answer
44 views

$R/Ra$ is an injective module over itself

Let $R$ be a PID, $a\in R$ be a nonzero nonunit in $R$. Prove that $R/Ra$ is an injective module over itself. If $R$ is a PID, every $R$- divisible module is injective, but the question concerns ...
0
votes
1answer
36 views

characterization of some finitely generated Z-modules

I want to characterize all finitely generated $\mathbb{Z}$-modules $M$ with the property that each submodule of $M$ is a direct summand of $M$. I think the module has to be torsion but I couldn't say ...