Questions tagged [modules]

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

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7 views

Direct sum of simple modules is cyclic

Let $R$ a ring, and $M$, $N$ simple $R$-modules non isomorph. Prove that $M\oplus N$ is cyclic. I am confused. If $(m,n)\neq 0$, $\phi: r\mapsto (m,n)r$ and the projections $\pi_i$ are R-homomorphism. ...
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2answers
15 views

Surjective linear transformation of finitely generated free modules and their determinants

I kind of got stuck on Aluffi Algebra Exercise 6.4 of chapter 6 which is: Let $F$ be a finitely generated free $R$-module and let $\alpha$ be a linear transformation of $F$. Give an example of an ...
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1answer
44 views

Projective finitely generated module over noetherian ring

Let $A$ be a noetherian ring and $M$ be a finitely generated $A$-module, I want to prove the following M is projective $\iff$ for all $P\subset A$ prime ideal the localisation $M_P$ is a free $A_P$-...
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13 views

What is an example of a Hilbert module that is not a Hilbert space?

Hilbert modules are always considered over some fixed C*-algebra of coefficients in which the generalised inner product takes values, so we cannot really say what a Hilbert module is without ...
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2answers
40 views

Show that $\text {Hom}_{\mathbb R[x]}(M,N)=\{0\}$ where $M,N=\mathbb R^2$ are $\mathbb R[x]$-modules where $X$ acts as $A$ in $M$ and $B$ in $N$

Let $A=\begin{pmatrix}2&0\\0&3\end{pmatrix}$ and $B=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ Show that $\text {Hom}_{\mathbb R[x]}(M,N)=\{0\}$ where $M,N=\mathbb R^2$ are $\mathbb R[X]$-...
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46 views

Dual of the Hom Space Between Modules

We know that if $V$ and $W$ are finite dimensional $\mathsf{k}$-vector spaces, then we have the isomorphism $$ \operatorname{Hom}_{\mathsf{k}}(V,W)^{*} \cong \operatorname{Hom}_{\mathsf{k}}(W,V). $$ ...
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1answer
16 views

Relation between R[X] and R[X,Y] as R-modules

In my lecture notes it was left as an exercise to show that for any $R$ $R[X,Y]\simeq R[X]$ as R-modules. To show this you need a bijective function $\phi$ such that $\phi (x+y)=\phi(x)+\phi(y)$ ...
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1answer
22 views

Example of torsion modules over integral domain that has zero annihilator

I was able to prove that if $R$ is an integral domain and $M$ is a finitely generated $R$-module is torsion if and only if $Ann(M) \neq 0$, but so far, I have failed to come up with an example that ...
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1answer
24 views

Is there exist a hollow module which is not Artinian

A module is said to be hollow if, its every proper submodule is small, that is, for any two proper submodule $N,K$ of $M$, $N+K\neq M$. A module is said to be Artinian if it satisfies descending chain ...
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5answers
77 views

Find the last two digit of $5^{121}*3^{312}$

The answers key says by using $$5^k\equiv 25 \pmod{100}, k>=2$$ and $$3^{40}\equiv 1 \pmod{100}$$ can have $$ 5^2 *3^4 \equiv25 \pmod{100},$$ and it follows that the last two digits of $5^{143}*3^{...
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1answer
74 views
+50

Reduced is equivalent to non-singular for commutative rings

Intro I've been (not successfully) trying to educate myself a bit about singular/nonsingular modules and I came by a proposition on Wikipedia that I find really interesting. For commutative rings, ...
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1answer
23 views

Submodules and $T$-invariant subspaces.

My professor wrote this on the board: Claim: there is a bijection between submodules and $T$-invariant subspaces {$T(w) \subset W$}. Proof: If $W \subset V$ is $T$-invariant, then $f(X).w = f(T)(w) \...
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20 views

Why isn't this statement about direct sums of injective modules contradicting the proposition?

I am trying to read Tsit-Yuen Lam's "Lectures on Modules and Rings" in order to understand why any abelian group can be injected in an injective abelian group, but I got confused by the ...
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28 views

understanding the contradiction of $I$ is not free.

My professor gave us this example on a module that is not free : $R = k[x,y,z]$ where $k$ is a field. $I = xyR + yz R + xz R \subset R.$ take $u = xy, v = yz, w = xz$ then $I = uR + vR + wR.$ And so, ...
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1answer
47 views

The proof that every $R$-basis of $M$ has the same number of elemets.

Here is the proposition my professor gave to us: Assume $R$ is commutative with unity. Let $M$ be a finitely generated free $R$- module. Then every basis of $M$ has the same no. of elelments called ...
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27 views

summand group ring [closed]

Let $D$ be a domain and $G$ be a finite group with $|G|^{- 1}\in D$. Assume that $DG=M\oplus N$ as $D$-module. Is there a way to convert $M$ as an $DG$-module direct summand of $DG$?
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Good filtrations on $A_n(K)$ modules

We are reading J. E. Björk's book: Rings of Differential Operators and we don't understand one step at Lemma 3.4: Let $\Gamma$ and $\Omega$ be two filtrations on the left $A_n(K)$-module $M$ and ...
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16 views

Module Over the Group Algebra of a Ring

I'm aware that if $K$ is a field, $V$ a $K$-vector space, and $G$ is a group, then a $K$-representation of $G$ in $V$ is the same thing as a $KG$-module, where $KG$ is the group algebra of $G$ over $K$...
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42 views

which one will best book for the following given below syllabus? [closed]

which one will best book for modules theory ? Jacobson or Dummit and footre ? Im confused about choosing the book which one will best book for the following given below syllabus ? Modules: ...
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1answer
21 views

Submodule over ring

I am calculating submodules of module $\mathbb{Z}_2[x]/(x^3+x+1)$ over $\mathbb{Z}_2[x]$. Consider $I=(x^3+x+1)$ ideal generated by $x^3+x+1$. Elements of $\mathbb{Z}_2[x]/(x^3+x+1)=\{0+I,1+I,x+I,x^2+...
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1answer
31 views

Given a ring with circumference n, if a person walks s distance per step on it, could we prove that the person will eventually step on every point?

If we add more restriction so that the question became Given a circle divided into x points each have same distance between each other. Now a person who walks s amount of point per step, if x and s ...
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1answer
43 views

What is the difference between finiteness for a module versus an algebra?

I encountered the following line on page 60 in section 1.5.3 of Shavarevich's Basic Algebraic Geometry I: "A ring $B$ that is finitely generated as an $A$-algebra is integral over $A$ if and only ...
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43 views

$k[x,y,z]$ and modules.

My professor gave us this example on modules (he started this by saying what is a basis? what is the meaning of linearly independent?): $R = k[x,y,z]$ where $k$ is a field. $I = xyR + yz R + xz R ,$ ...
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1answer
24 views

Surjective map between modules and generators

Suppose that $ \phi : N \to M$ is a surjective $R$-linear map. Show that if $N$ is finitely generated then so is $M$; and on the other hand if $M$ and ker $\phi$ are finitely generated then $N$ is ...
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1answer
22 views

Vector space and $\mathbb Z_n.$

My professor gave us this as an example of a module (I guess my professor were comparing the linear independence in a vector space to that in a module): In a vector space $V$ over a field $k,$ we have ...
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2answers
94 views

Integer-valued polynomials divisible by 2 or 3

I would like some help with the following problem: Suppose that $p \in \operatorname{Int}(\mathbb{Z}) = \{p \in \mathbb{Q}[X] : p(\mathbb{Z}) \subseteq \mathbb{Z}\}$ is such that for all $z \in \...
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10 views

Homogeneous elements of graded submodule

Let $R$ be a graded ring, $N$ and $M$ be graded $R$-modules. Then, $N$ is a graded submodule of $M$ if and only if $N$ is generated by the homogeneous elements of $M$ which belong to $N$. I proved ...
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1answer
23 views

Length of submodule

Does there exists a module $M$ of a finite length(greater than $3$) over a ring $R$ such that each submodule of $M$ has different length?(not a simple module) Let $M$ be a module and $0=M_0\subset M_1 ...
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1answer
64 views

Laurent polynomials $ \mathbb C[t,t^{-1}]$ is the localization of $\mathbb {C}[t].$

I want to prove this question: Show that the ring of Laurent polynomials $ \mathbb C[t,t^{-1}]$ is the localization of the polynomial ring $\mathbb {C}[t].$ Localization is defined as follows: Let $R$ ...
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35 views

Proof that $\operatorname{Ext}^n_{kG}(M,N) = \operatorname{Hom}(\Omega^nM, N)$

Let $G$ be a finite group and let $k$ be a field of modular characteristic. Write $\mathsf{St}_k(G)$ for the stable module category of $G$. It is the triangulated category obtained from $\mathsf{Mod}...
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2answers
50 views

Homomorphism of $R$-modules gives a commutative diagram

I am working on Problem 2.10 of Atiyah & Macdonald's Introduction to Commutative Algebra. In this problem, we have a commutative ring $A$ and a homomorphism $u : M \longrightarrow N$ of $A$-...
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1answer
31 views

Reflexive modules and the canonical map $M\to \operatorname{Hom}_R(\operatorname{Hom}_R(M,N),N)$

Let $R$ be a commutative Noetherian ring. For any $R$-module $M,N$, there exists a canonical map $d_{M,N}: M \to \operatorname{Hom}_R(\operatorname{Hom}_R(M,N),N)$ sending an element $m \in M$ to the ...
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0answers
32 views

presentation of a generic algebra

In the book $\ulcorner$Reflection Groups and Coxeter Groups$\lrcorner$ written by J. E. Humphreys, in the beginning of chapter 7 $<$Hecke algebras and Kazhdan-Lusztig polynomials$>$, it defines ...
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1answer
37 views

z complex $p, q \in \mathbb{N}, p < q$ such that $|z ^ p + \frac {1}{z ^ p}| \ge |z ^ q + \frac {1}{z ^ q}|$, so $|z + \frac{1}{z}| < 2$

I have participated yesterday in a contest. One of the problems was this one: If $x \in \mathbb{C} \setminus \mathbb{R}$ and there exists $p, q \in \mathbb{N}, p < q$ such that $|z ^ p + \frac {1}{...
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1answer
61 views

Injective Modules Motivation & Intuition

A module $M$ over a commutative ring $R$ is called a 'injective module' if it satisfies certain universal property explaned here. Question: Is there any intuition how to think concretely about ...
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2answers
37 views

linearly isomorphic submodule

I'm currently learning about modules and am pretty new to the topic. I recently found this question: Suppose that $R$ is an integral domain and not a field. Give, with proof, an eample of a proper ...
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0answers
40 views

module isomorphisms

so I am currently learning about modules, I'm pretty new to them, but i have some experience with rings and linear algebra and stuff. I got the following problem: Let $R$ be the ring $\mathbb{Z}[\...
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1answer
44 views

Understanding why $B$ is generated by a finite number of monomials and how this affects $B_0$.

Here is the question I want to understand its statement: Let $B = \mathbb C[x_1, x_2, x_3, x_4]$ be $\mathbb Z$-graded so that each $x_i$ is homogeneous and $\deg(x_1, x_2, x_3, x_4) = (1, -2, 3, -4)....
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25 views

Direct limit; what is the conditions for canonical homomorphism to be surjective

I am studying directed system and direct limit. Consider a directed system $(A_{i},\alpha_{i})$, where $A_{i}$ is a sequence of modules(or algebras, groups,etc). $\alpha_{i}:A_{i}\rightarrow A_{i+1}$ ...
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53 views

When $M^{**} \cong M$ [duplicate]

Let $R$ be a ring, and let $M$ be an $R$-module. Can we characterize when the canonical map $M \rightarrow [[M, R]_{R \text{-mod}} , R ]_{R \text{-mod}}$ sending $a$ to $\hat{a} : [M, R]_{R \text{-mod}...
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1answer
67 views

Understanding Cayley-Hamilton Theorem for Modules

I'm trying to understand the Cayley-Hamilton Theorem for modules, here is what I'm having trouble with. PROPOSITION (Cayley-Hamilton) Suppose $I \subseteq A$ is an ideal and $\phi:M→M$ is an A-module ...
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1answer
25 views

When does injective implies surjective for an $R$-module endomorphism?

Let $R$ be a commutative ring. It is known that if $M$ is a finitely generated $R$-module and if $f \in End_R(M)$, then surjectivity of $f$ implies injectivity of $f$ (Vasconcelos). I am looking for ...
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1answer
54 views

Isomorphic as $R$-modules v.s. Isomorphic as abelian groups

Let $(R,\mathfrak m, \mathbb Q)$ be a Noetherian local ring. Let $M$ be a finitely generated $R$-module such that for some integer $n\ge 0$, there is an isomorphism of abelian groups $M \cong \mathbb ...
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1answer
25 views

$M\subset N$ for $R$-modules $M,N$ if $S_{\mathfrak m}^{-1}M\subset S_{\mathfrak m}^{-1}N$ for all maximal ideals $\mathfrak m\subset R$?

Consider the following proposition (with proof) taken from S. Lang's "Algebraic Number Theory": Proposition $\mathbf{18}$. Let $A$ be a Dedekind domain and $M,N$ two modules over $A$. If $\...
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0answers
58 views

$\text{Hom}_R(R,M) \cong M$? [duplicate]

My question is as simple as the title. Is it true that $\text{Hom}_R(R,M)$ is isomorphic to $M$? In which $R$ is a commutative ring with unit, and $M$ is an $R$-module. I actually made a proof, I just ...
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1answer
38 views

the proof of faithfully flat module is faithful

I know faithful and flat module is not always faithfully flat. For example $\mathbb Q \otimes_{\mathbb Z} \mathbb Z/2\mathbb Z=0$ implies $\mathbb Q$ is not faithfully flat as $\mathbb Z$-module ...
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1answer
52 views

Equivalence of $R$-linear additive categories are $R$-linear?

Let $R$ be a commutative ring. Let $C,D$ be $R$-linear categories (https://stacks.math.columbia.edu/tag/09MJ ) such that they are also additive categories. Let $F: C \to D$ be an equivalence of ...
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0answers
26 views

Homomorphisms of quotient field into domain

Let $R$ be an integral domain that is not a field, with quotient field $\operatorname{Quot} R$. We view both as $R$-modules. I want to show that there is no non-trivial $R$-module homomorphism $\phi: \...
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0answers
23 views

Maximizing the rank of a 0-1 matrix over a field vs. a ring

In one my classes, we proved that the rank of a 0-1 matrix (i.e. a matrix whose entries are all 0 or 1) is maximized over $\mathbb{Q}$. More precisely, because every field has $0$ and $1$, we may ...
2
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1answer
34 views

Understanding how the induced representation of $S_3$ under $S_2$ with the standard representation works

I am attempting to understand the representation of the induced module $\mathrm{Ind}_{S_2}^{S_3}V = \Bbbk S_3 \otimes_{\Bbbk S_2} V$, where $V = \{v \in \Bbbk^2: v_1+v_2=0\} = \Bbbk(e_1 - e_2)$, and ...

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