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

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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: ...
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1answer
8 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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33 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
39 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
28 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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15 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
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1answer
36 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
31 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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$P$ 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
29 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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34 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
32 views

Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2 , \cdots x_n \cdots$ (Of course, each element of $F$, being a polynomial, will involve only ...
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2answers
50 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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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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55 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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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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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 ...
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3answers
43 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 ...
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20 views

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

$e_A(\varphi,M) = l_A(\mathrm{coker}\varphi) - l_A(\ker\varphi)$ 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 ...
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23 views

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

the primary for the exercise idea is to use the universal property of the external direct sum of modules.
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1answer
67 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 ...
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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 ...
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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 ...
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41 views

Factorization in modular arithmetic

Is this expansion a legal step? $12^8\mod15 = 2^8 * 2 ^ 8 * 3 ^ 8\mod15$
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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 ...
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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 ...
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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 ...
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1answer
34 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 ...
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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 ...
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34 views

When does a real inverrtible matrix send a lattice in $\mathbb{R}^2$ to itself?

There is a question in Artin asking when does an inverrtible 2x2 real matrix send a lattice to itself. The issue I have is with this question not being specific as to what kind of answer is ...
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37 views

When do three complex numbers define a lattice?

Given three complex numbers, when is the set of all integer linear combinations of them a lattice? Since lattices can be defined by only two basis elements, I suppose the answer must be that one ...
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1answer
46 views

Subring of a commutative Noetherian ring

We know that it's possible subring of the commutative Noetherian ring become not Noetherian (for example: Subring of a Noetherian ring need not be Noetherian?). But if $S$ be a subring of ...
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1answer
95 views

Is this module injective? [duplicate]

Here's the problem: (This problem is from Hungerford's Algebra Chapter 5 exercise 6.7 ) Let R be a principal ideal domain and $p$ a prime in $R$ and $n$ a positive integer. Then $R/(p^{n})$ is an ...
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1answer
21 views

Module isomorphism from $R$ to $R \oplus R$ for a certain ring $R$

My textbook says: Let $R$ denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. Show that $R \cong R \oplus R$ as $R$–modules. So for $A, B \in R$, I tried ...
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2answers
43 views

What are the semisimple $\mathbb{Z}$-modules?

What are the semisimple $\mathbb{Z}$-modules? Comments: I think they are direct sums of copies of such $\mathbb{Z}_p$'s, where $p$ is a prime number. I believe it is, but I can not prove.
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1answer
31 views

generating system of the kernel of a module-transformation

Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module. Next, let's consider the transformation: $$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g ...
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35 views

Show that if $M$ is a semisimple artinian module then $M$ is finitely generated.

The exercise is as follows: Show that for a semisimple module $M$ over any ring, the following conditions are equivalent: $(1)$ $M$ is finitely generated; $(2)$ $M$ is Noetherian; $(3)$ $M$ is ...
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51 views

If R is a Noetherian ring, is it true that there exists an integer N such that every ideal of R is generated by at most N elements? [duplicate]

I think, as R is an ideal of itself, and R is Noetherian thus R is finite generated, assume it is generated by N elements, then every ideal is generated by at most N elements. But my conclusion is ...
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22 views

Constructing a module homomorphism

Let $\phi: M \to M'$ be a homomorphism of left R-modules, and let $N' \subset M'$ be a submodule. Construct a natural module homomorphism: $\tilde{\phi}: M/\phi^{-1}(N') \to M'/N'$ and show that ...
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2answers
22 views

$\displaystyle \mathbb KG\simeq \frac{\mathbb K[t]}{\langle t^n-1\rangle}$ as algebras?

Suppose $G$ is a cyclic group of order $n<\infty$. How can I show the isomorphism of algebras: $$\mathbb KG\simeq \frac{\mathbb K[t]}{\langle t^n-1\rangle}?$$ Above $\mathbb K$ is a field, ...
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39 views

Colimits in $Ch_R$, help with a step of the proof

I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same ...
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26 views

Number rings as free module over base ring

Let $K \subset L$ be number fields and $\mathcal{O_K}, \mathcal{O_L}$ the corresponding rings of algebraic integers. Further let dimension$(L/K)$ = n as a vector space. If $K$ is a PID, then ...
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26 views

Finite tensor product of infinite $R$-modules

I was challenged to find a ring $R$ and infinite $R$-modules $A$ and $B$ such that $A \otimes_R B$ is a finite but nontrivial abelian group. I can think of many examples that give $0$, but I'm ...
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21 views

Regard naturally as modules.

Suppose that $I$ is a two-sided ideal in the ring $R$, and that $M$ is a module over the quotient ring $R/I$. Why can we naturally regard $M$ as a $R$-module that is annihilated by $I$? Conversely, ...
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58 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
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11 views

Is the following short exact sequence always split?

Is it always true that the following short exact sequence of modules split? $0\to N\to M\to M/N\to0$
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1answer
42 views

Does the bicategory of bimodules really have left Kan extensions?

Let $\mathsf{Bimod}$ denote the bicategory whose $0$-cells are rings $A$, a $1$-cell $A \longrightarrow B$ is an $(A,B)$-bimodule and a $2$-cell is a bimodule map. The composition of $1$-cells is ...