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Questions tagged [modules]

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

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Is the tensor product of a free module with any other module free? [closed]

Let $A$ be a ring and $F, M$ two $A$-modules such that $F$ is free. Is then the tensor product $M \otimes_A F$ free? I know that the tensor product is free if both of them are free $A$-modules. But ...
ATW's user avatar
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Why $eAe$ is identical with $\mathrm{End}(eA)$ [duplicate]

Let $A$ be an algebra and $e$ is an (maybe full) idempotent. I saw there is an isomophism between $eAe$ and $\mathrm{End}(eA)$. How is it constructed?
Sirin's user avatar
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1 answer
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Tower of module-endomorphism rings

Let $R$ be a ring with nonzero left ideal $A$. Define $E_1=\text{End}({}_RA)$ viewed as a ring of right operators on $A$ and $E_2=\text{End}(A_{E_1})$ viewed as a ring of left operators on $A$. ...
khashayar's user avatar
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1 vote
1 answer
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Double Centralizer Property in simple ring without identity

In Lam's book, $\textit{A first course in noncommutative rings}$, Theorem 3.11 states: Let $R$ be a simple ring, and $A$ be a nonzero left ideal. Let $D = End({}_RA)$ (viewed as a ring of right ...
khashayar's user avatar
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2 votes
0 answers
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singular ideal of matrix rings

Let $R$ be a ring with identity. An element $x\in R$ is said to be right singular if $xI=0$ for some essential right ideal $I$ of $R$. The set of all right singular elements of $R$ is denoted by $Z_r(...
Dr. Nirbhay Kumar's user avatar
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47 views

What is $A\otimes_k (A/k)$?

Let $A$ is an associative algebra over a field $k$, I want to understand $A\otimes_k (A/k)$. Parenthesis matter (as this is often written in terms of $\bar A=A/k$). I'm happy if I know this for $k=\...
c.p.'s user avatar
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rational functions flat over $k[x]$

I was looking at the solution of an exercise, which proved the following statement: The $k[x]$-module $k(x)$ of rational functions for $k$ a field is flat. The solution goes as follows: Let $\phi: N ...
Pastudent's user avatar
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Flat Module from book of NS Gopala Krishnan

I was reading flat modules from book by NS Gopalakrishnan. In starting of faithfully flat algebra the below is written Let $A$ be an $R$-algebra, $M$ and $N$ are $R$-modules. Then homomorphism $\phi_M ...
Swaraj Koley's user avatar
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26 views

First syzygy is unique up to direct summand

Let $R:=K[x_0,...,x_n]$ be the polynomial ring over a field $K$ and $M$ be finitely generated $R$ module. Let $(m_i)_{i=1,...,k}$ be a generating set for $M$. Then we can define the free module $F_0 :=...
Flynn Fehre's user avatar
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Length and complete decomposability of $(\varphi,\Gamma)$-modules

I am following the following book on Galois representations and $(\varphi,\Gamma)$-modules. Let $p$ be prime and $L$ a finite $\mathbb{Q}_p$ extension. Let $\Gamma= \mathrm{Gal}(L_\infty/L)$ be the ...
farik-amin's user avatar
1 vote
2 answers
57 views

Find the natural number n for which the set $A_n$ has exactly 323 integers

the problem For n a natural number we define $A_n=\{x\in \Bbb R \,|\,\, |x+n+4| \leq 3n-4\}$. The natural number n for which the set $A_n$ has exactly 323 integers is....? my idea i absolutely have ...
IONELA BUCIU's user avatar
2 votes
0 answers
68 views

$\varphi$-modules that are not completely decomposable

Forgive me for I am not an expert in the area, and I fear that this may be a somewhat trivial question - so thank you for your patience! I am reading Brinon-Conrad's notes on $p$-adic Hodge theoy, ...
mathieu_matheux's user avatar
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Connection between twisted graded modules and twisted sheaves

I came across the definitions of graded twisted modules while reading about the syzygy theorem. In the meantime I also attended an algebraic geometry course where twisted sheaves occured. For both ...
Flynn Fehre's user avatar
1 vote
1 answer
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Number of left ideals in the simple components of groups algebras

Let $G$ be a finite group and $K$ a field with characteristic zero. Suppose $G$ has $m$ irreducible $K$-representation $W_i$ with character $\chi_i$. $KG$ is semisimple algebra, and $$KG=KGe_1 \times \...
khashayar's user avatar
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The residue class of a complete set of primitive orthogonal idempotents [closed]

I was studying the book Elements of the Representation Theory of Associative Algebras: Volume 1 and this question occurred to me: In page 29, it says that Because $\{e_1,…,e_n\}$ is a complete set of ...
Chestnuto's user avatar
1 vote
1 answer
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Let $R$ be a ring and $M$ a (left) $R$-module.

Let $R$ be a ring and $M$ a (left) $R$-module. (a) Prove that for every $z \in Z(R)$ the set $zM = \{ zm \mid m \in M \}$ is a submodule of $M$. (b) Give an example of a ring $R$ and an idempotent $e \...
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Let $\varphi: K_1 \rightarrow K_2$ be a ring homomorphism and $M$ a left $K_2$-module.

Let $\varphi: K_1 \rightarrow K_2$ be a ring homomorphism and $M$ a left $K_2$-module. (a) Prove that $M$ becomes a left $K_1$-module if we introduce the operation of multiplication by elements from $...
Markus's user avatar
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Free module and finitely generated submodule.

Let $A$ be a commutative ring, $M$ a free $A$-module and $N\subseteq M$ a finitely generated submodule. Prove that there is a Sub-Noetherian Ring $A_0 \subseteq A$, a free $A_0$-submodule $M_0 \...
George's user avatar
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1 vote
1 answer
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Questions on the $\hom$ Functor and Free Groups

This question arises while learning about the $\hom$ functor. My algebra background is not that strong, so here is my question: Let $G$ be a free group, and let $f\colon G \to G$ be a group ...
Random's user avatar
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5 votes
1 answer
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Finiteness of a generalization of the class number of a number field

Let $f(x)$ be an irreducible monic integral polynomial with root $\alpha \in \mathbb{C}$. A classical result of Latimer and MacDuffee asserts that there is a bijection between similarity classes of ...
Ben Marlin's user avatar
1 vote
1 answer
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Cohomology group for trivial group

Let $G$ be a group, $A, B$ be a $G$ - modulo. We can define the n-th Cohomology group of $G$ with coefficient in $A$. $$H^n(G,A) =\text{Ext }_G^n(\mathbb{Z},A)$$ And the n-th Homology group of $G$ ...
Kongca's user avatar
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Motivation for Tensor Product of Modules

How did the motivation and background for the definition of the tensor product in modules arise, and what was the process of abstraction that led to the formation of the definition of the tensor ...
Yassin Dwi Cahyo's user avatar
1 vote
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Set of primitive idempotents of group algebras

Let $G$ be a finite group and $K$ be a field with characteristic zero. Can we construct the set of primitive orthogonal idempotents of $KG$? By this set, I mean the set of idempotents such that $...
khashayar's user avatar
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2 votes
1 answer
66 views

For two irreducible modules $V$ and $W$, $f : V\to W$ is a $G$-module isomorphism $\iff \text{span}(f)\subset V^*\otimes W$ is trivial

I 'm self-studying some representation theory and I was trying to solve a problem that says Show that if $V$ and $W$ are irreducible G-modules, then $f : V\to W$ is a $G$-module isomorphism if and ...
Fung San Gaan's user avatar
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38 views

Showing irreducibility of representations and modules

Let $\rho:S_5 \rightarrow GL_5(\mathbb{C})$ be the representation of the permutation matrix $(e_{\sigma(1)},...,e_{\sigma(5)})$. I am stuck on the following: Why is $U= \{a_1e_1+...+a_5e_5|a_1+...+...
Very Interesting's user avatar
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Is $\bigoplus_{n \in \mathbb{N}}\mathbb{Q}$ reflexive?

It is well known that $\text{Hom}_{\mathbb{Z}}(\prod_{n\in \mathbb{N}}\mathbb{Z},\mathbb{Z}) = \bigoplus_{n \in \mathbb{N}} \mathbb{Z}$ holds. From this result, it can be shown that $\bigoplus_{n \in \...
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0 answers
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Does it make sense to extend the definition of the ideal quotient to modules?

So, just for a bit of context, I'm trying to learn commutative algebra and right now I'm trying to look at interesting operations for combining ideals to make new ideals. I don't have an understanding ...
Greg Nisbet's user avatar
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2 votes
1 answer
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What is the relation between the tensor of two $R$-modules over different rings. [closed]

I'm new to tensor products and I stumbled upon the following four short, related questions during self-study (let us assume that $R$ is a commutative ring with unity): Let $M$ and $N$ be $R$-modules. ...
ShamanR's user avatar
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If the graded module is finitely generated then the filtration is good

Suppose $A$ is a unital Noetherian ring with an ideal $\mathfrak{q}$. Provide $A$ with its $\mathfrak{q}$-adic filtration. Let $M$ be a finitely generated $A$-module with descending filtration $(M_n)$ ...
ephe's user avatar
  • 446
-1 votes
2 answers
50 views

Adjoints of linear maps between modules

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules. Let $\langle \cdot, \cdot \rangle: M \times N \to R$ be a bilinear form. Let $f: M \to M$ be an $R$-Linear map. Does there necessarily ...
Smiley1000's user avatar
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0 answers
16 views

Primitive non-central idempotents of a group algebra

Let $W_i;\ 1 \le i \le m$ be irreducible $\mathbb{C}$-representations of a finite group $G$ with $\mathbb{C}$-characters $\chi_i$. Let $(V,\rho$) be a $\mathbb{C}$-representation of $G$ with isotypic ...
khashayar's user avatar
  • 2,331
6 votes
1 answer
83 views

Schur’s lemma over $\mathbb{F}_p$

I’m studying modular representation theory, and I got really stuck with the seemingly innocent statement. Consider $\mathrm{GL}_{2}(\mathbb{F}_{p})$ and its center $Z$, which is just a set of all ...
Matthew Willow's user avatar
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0 answers
19 views

Module isomorphism of tensor products

Let $A$ be a $K$-algebra, and $M$ be a simple $A$-module. Is there any isomorphism between $\text{Hom}_A(M,M^n) \otimes_K M$ and $\text{End}_A(M)\otimes_K M^n$? Can we conclude they are module ...
khashayar's user avatar
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1 vote
1 answer
90 views

Does every ring have a set's worth of simple modules?

Over certain kinds of rings, i.e. Artinian rings, there are only finitely many simple modules up to isomorphism. What can we say about simple modules over general rings? In particular, do the ...
Jackson Wilson's user avatar
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0 answers
24 views

Decomposition of primitive central idempotents in group algebras

Let $W$ be an irreducible $\mathbb{C}$-representation of a finite group $G$ with character $\chi_W$. A primitive central idempotents of the group algebra $CG$ is: $$e=\frac{\dim_{\mathbb{C}}(W)}{|G|}\...
khashayar's user avatar
  • 2,331
2 votes
2 answers
179 views

Endomorphisms on certain simple modules are multiples of the identity

Let $R$ be a (not necessarily commutative) ring with unit and $M$ be a simple left $R$-module. If, furthermore, $R$ is an algebra over $\mathbb{C}$, then every endomorphism on $M$ is a multiple of the ...
SilverBladeII's user avatar
4 votes
0 answers
31 views

Isomorphism between Groupring on complex numbers and direct sum of irreducible representations

For a group G we get irreducible representations $\phi:G \rightarrow GL(V_i)$. I want to see why $ \mathbb{C}G \cong \bigoplus\limits_{i=1,...,r}End(V_i)$. I have been looking at the proof of ...
LostInTheSauce's user avatar
1 vote
0 answers
35 views

Decomposing a certain kind of isometry of modules

Let $R$ be a finite ring, $R_1 \times \dots \times R_m$ its decomposition into finite local rings. Let $\pi_i$ denote the projection to each factor $R_i$. Then $\pi_i$ extends componentwise to a map $\...
JBuck's user avatar
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1 vote
1 answer
48 views

When are Idempotents elements of a semisimple algebra primitive

Let $A=KG$ be a $K$-algebra such that $|G| \in K^{\times}$. Here $A$ is a semisimple algebra. Consider the decomposition of $A$ into simple components:$$A=A_1 \times A_2 \times \cdots \times A_k.$$ ...
khashayar's user avatar
  • 2,331
2 votes
4 answers
104 views

Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

in my algebra class, they give us as an exercise to prove that a finite group $G$ admits at most finitely many non-equivalent irreducible representations over an arbitrary field $F$. Now, I showed ...
F. Salviati's user avatar
5 votes
1 answer
110 views

Showing a matrix is not similar to its transpose over $\mathbb Z$

This answer claims the matrix $A = \begin{bmatrix}8 & 2 \\ 0 & 1\end{bmatrix}$ is not similar over $\mathbb Z$ to its transpose, without telling why. By taking a generic $S = \begin{bmatrix}x &...
Carla_'s user avatar
  • 457
0 votes
0 answers
20 views

Th substitution property of modules implies the cancellation property of modules

We say that a module $M_R$ satisfies the internal cancellation property (or $M$ is internally cancellable) if whenever $M=A\oplus X = B\oplus Y$ with $A \cong B$, then $X \cong Y$. We say that $M$ ...
Hussein Eid's user avatar
  • 1,071
1 vote
0 answers
26 views

Primitive idempotent and bilateral ideals

I'm trying to show for my algebra class that in a semisimple ring with unity $R$ (not necessarily commutative), every primitive idempotent element must belong to a minimal two-sided ideal. Here, by ...
F. Salviati's user avatar
3 votes
1 answer
252 views

Faithfulness of a module: is it a categorical property?

Let $R$ be a (unital) ring and let $M$ be a left $R$-module. The $R$-module $M$ is called faithful if $$rM = 0\implies r = 0$$ holds for every $r\in R$, i.e. the left ideal $$\operatorname{Ann}_R(M)=\{...
Andromeda's user avatar
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0 votes
0 answers
99 views

Compatibility - Topological modules contra vector spaces

So Tréves in his book on topological vector spaces shows that a filter $\mathcal{F}$ on a $\textit{vector space}$ $E$ is the filter of neighbourhoods of zero compatible with the linear structure if ...
undefined's user avatar
  • 277
2 votes
0 answers
52 views

$\operatorname{Hom}_k(R,k)$ is injective indecomposable $R$-module for $R$ local $k$-algebra of finite $k$-dimension

I am working on Exercise 3.1.22. from Bruns and Herzog's Cohen-Macaulay rings. The exercise in question is the following (the references I will use are of course from within the book): It appears ...
Joel Castillo Rey's user avatar
1 vote
1 answer
64 views

$\operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \otimes_R S$?

Let $R$ and $S$ be commutative rings with unity. Let $f: R \to S$ be a ring homomorphism. Let $M$ be an $R$-module. Do we have $$ \operatorname{End}_S(M \otimes_R S) \cong \operatorname{End}_R(M) \...
Smiley1000's user avatar
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0 votes
2 answers
41 views

Relationship Between Subgroups of Abelian Groups & Ideals/Rings.

Clarification. I am currently reading from Dummit and Foote. Given $R$-module $M,$ we require $(1)$ $R$ is unital, and $(2)$ $1\cdot x=x$ for all $x\in M.$ When discussing rings $R,$ for the purposes ...
JAG131's user avatar
  • 917
2 votes
0 answers
67 views

Does reverse homology $\ker \delta\subset \operatorname{im} \delta$ have a defining equation? I don't think it does.

Background Material. Is "reverse homology" $\ker g \subset \text{im} f$ possible? Question. Forward (usual / historical) homology always has the defining equation $\delta^2 = 0$ for a ...
SeekingAMathGeekGirlfriend's user avatar
0 votes
1 answer
20 views

Canonical form of presentation matrix

Setup: We know that an $R$-module $M$ is finally presented if it is isomorphic to $R^m/AR^n$, where $A$ is an $m$-by-$n$ matrix known as the presentation matrix. Determining $A$ amounts to picking a ...
Damalone's user avatar
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