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

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16 views

Cancellation of finitely generated modules over a PID

Suppose that $A=\mathbb R[x]$, $D = A/⟨x^2+1⟩⊕A^2$. $B$ and $C$ are finitely generated $A$-modules. Suppose that $D⊕B \cong D ⊕ C$. How to show $B\cong C$? I tried to solve it but I have no clue, ...
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2answers
23 views

Why is a submodule of a free module over a PID is free?

Rotman - Advanced modern algebra p.650 Theorem 9.8 Let $R$ be a PID and $M$ be a free $R$-module and $N$ be an $R$-submodule of $M$. Let $\beta$ be an $R$-basis for $M$ and well-order it. Now, ...
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1answer
44 views

relation between vector space and torsion free module [on hold]

Can you please help me to prove this. Let $R$ be a domain, $A$ be an $R$-module, and $Q=Frac(R)$. Then a module $A$ is a vector space over $Q$ if and only if it is torsion-free and divisible. ...
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3answers
26 views

$||x-2|-3| >1$, then $x$ belongs to which interval

$||x-2|-3| >1$, then $x$ belongs to: (a) $(-\infty, -2) \cup (0,4) \cup(6,\infty)$ (b) $(-1,1)$ (c) $(-\infty, 1)\cup (1, > \infty)$ (d) $(-2,2)$ Answer:(a) My solution: let $|x-2| = p,$ ...
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1answer
17 views

The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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1answer
17 views

Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
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2answers
28 views

If $I$ and $J$ are ideals of an $R$-algebra $A=I+J$ then $I\oplus J\simeq A\oplus (I\cap J)$?

Let $I$ and $J$ be two left ideals of an $R$-algebra $A$ such that $A=I+J$. Here $R$ is commutative ring with identity $1_R$. How can I show $$I\oplus J\simeq A\oplus (I\cap J)?$$ I've tried several ...
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0answers
11 views

Injective homomorphism on tensor product

I am currently attempting the following: Find (cyclic) $\mathbb{Z}$-modules $M, N, P$ and an injective homomorphism $f: M \rightarrow N$ s.t. $g: M \otimes_{\mathbb{Z}} P \rightarrow N ...
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0answers
20 views

Submodules $H$ satisfying: “if $ax \in H$ for some non-zero scalar $a$, then $x \in H$.”

Suppose $R$ is a commutative ring and that $X$ is an $R$-module. Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following? For all $x \in X$, if $ax \in H$ ...
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2answers
19 views

Is there another terminology to designate this?

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Then there exists a free $R$-submodule $F$ of $M$ such that $M=Tor(M)\oplus F$ and the ranks of such $F$'s are the ...
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1answer
13 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 ...
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0answers
21 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 ...
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24 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) $\text{Ann}_A(N)$ ...
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22 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. ...
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31 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 ...
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2answers
42 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?
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4answers
117 views

Is there a counterexample for the claim: if $A \oplus B\cong A\oplus C$ then $B\cong C$?

Here $A$, $B$ and $C$ are $R$-modules. Is there a counterexample for the claim: if $A \oplus B\cong A\oplus C$ then $B\cong C$? And what if $B$ and $C$ are finitely generated?
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0answers
35 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
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1answer
40 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 ...
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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 ...
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1answer
25 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 ...
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1answer
24 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 $\mathbb Z[\omega]$? ...
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22 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
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1answer
18 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
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2answers
47 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 ...
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1answer
35 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.
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1answer
14 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 ...
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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
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1answer
69 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 ...
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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 ...
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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 ...
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0answers
41 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 $IJ = I \cap J$, ...
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1answer
42 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 ...
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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 ...
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1answer
30 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 ...
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0answers
17 views

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

I want an example to show that infinite direct sum of injective modules is not injective
2
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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 ...
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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 ...
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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 ...
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1answer
33 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 + ...
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0answers
41 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 ...
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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 ...
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2answers
54 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...