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

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equivalence of categories between modules and vector space

Let $Map_k$ the category where the objects are triples $(V,W,f)$, where $V$ and $W$ are finite dimensional $K$-vector spaces and $f:V\rightarrow W$ is a $K$-linear map. A morphism from $(V,W,f)$ to $(...
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0answers
18 views

free modules being isomorphic with different dimension

This question is from the book A course in Homological algebra by Hilton and Stammbach Let $V$ be a vector space of countable dimension over the field $K$. Let $\Lambda=\rm{Hom}_K(V,V)$. show that, ...
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1answer
24 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely genereated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and ...
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1answer
53 views

Is there a theory of generalized eigenvectors over commutative rings?

Brown's Matrices over Commutative Rings book discusses the theory of eigenvalues, eigenvectors, and diagonalizing matrices over commutative rings, but unless I've missed something, nothing like ...
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1answer
23 views

Property of free modules

Question is regarding following property of free modules: Let $P$ be a free $R$ module. To every surjective homomorphism $f:B\rightarrow C$ of $R$ modules and to every homomorphism $g:P\...
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3answers
40 views

Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
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2answers
26 views

Help with proving a fact about injective modules

Let $R$ be a ring with identity. I am trying to show that an $R$-module $A$ is injective if and and only if for every left ideal $L$ of $R$ and every $R$-module homomorphism $g: L \rightarrow A$, ...
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1answer
27 views

A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
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0answers
33 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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0answers
10 views

Indecomposable Modules over an algebra

If I have $K$ subfield of $F$ and $A$ an $K-$algebra (associative ,finite dimensional and with with unity). And take an ciclic indecomposable $A-$module $M=A\lambda$. Then $A\otimes_K F(\lambda\...
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1answer
41 views

Isomorphic for free modules implies for projective [on hold]

I have if $F$ is a free $R$-module then $\mathrm{Hom}(L,M) \otimes F$ is isomorphic to $\mathrm{Hom}(L,M \otimes F)$. Then how can I conclude that $\mathrm{Hom}(L,M) \otimes P$ is isomorphic to $\...
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27 views

finitely generated projective R-modules

Let $L$, $M$ and $N$ be $R$-modules. Then I know that there is a natural homomorphism from $Hom(L,M) \otimes N \to Hom(L,M \otimes N)$ defined by $f \otimes n \to \tilde{f}$ where $\tilde{f}(\ell)=f(\...
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0answers
20 views

What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...
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0answers
14 views

Application of Structure Theorem to Prove Simultaneous Diagonalizability and Group of Units of Cyclic Groups

I am reading these notes on Modules over PID. Exercise 67 (pg 24) asks to prove that: Problem. Let $A$ and $B$ be $n\times n$ matrices with complex entries. Then $A$ and $B$ are simultaneously ...
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1answer
34 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
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1answer
22 views

Polynomial ring as direct sum of modules

I'm aware that this topic has been posted previously. However, I'm not wanting to prove anything. I just want to understand, somewhat intuitively, why $R[x]$ as a module over $R$ is given by $$R[x] =\...
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1answer
30 views

The sum of a family of submodules

I'm just reading Atiyah-Macdonald's commutative algebra text, specifically the section on Operations on Submodules. I just have some questions regarding constructions. Firstly, suppose $M$ is an $A$-...
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1answer
49 views

Is it $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$ true? [closed]

Suppose that $A$ is a ring and $S$ is a multiplicative closed subset of $A$. I wonder that do we have $\prod_{i \in \Delta } (S^{-1}A_{i}) \cong S^{-1} \prod_{i \in \Delta }A_{i}$? Can you give me ...
2
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1answer
29 views

What is the simplest example of the tame representation type?

What is the simplest example of the tame representation type? I tried to find simple example could help me to understand the tame representation type. I know the definition of tame is like: A ...
2
votes
1answer
31 views

Hom and direct sums 3

Let $\{M_i\}_{i \in I}$ and N be left R modules where R is not necessarily commutative. Then how can we prove that $Hom_R(N, \bigoplus_{i \in I} M_i)$ is isomorphic to $\bigoplus_{i \in I}Hom_R(N,M_i)...
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0answers
21 views

If $N$ is finitely generated, then $(L :_{R} N)^{e}= (S^{-1}L :_{S^{-1}R} S^{-1}N)$ [duplicate]

I have question in this lemma. Please help me explain it more. Let $L$, $N$ be submodules of a module $M$ over a commutative ring $R$ and let $S$ be a multiplicatively closed subset of $R$. If $N$ is ...
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2answers
42 views

How many solutions of $\mod 63$ : $x^2=1 \pmod7 $ and $x^3=1\pmod 9$ [closed]

How many solutions $\mod 63$ , we have for: $$x^2=1 \pmod 7$$ and $$x^3=1 \pmod 9$$ Need to find them also.
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0answers
24 views

Trivial extension of an opposite algebra

Suppose that $A$ is a finite dimensional $K$-algebra, where $K$ denote an algebraically closed field. Call $DA=Hom_k(A,k)$. $DA$ admits an $A$-$A$-bimodule structure in the obvious way. The trivial ...
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2answers
25 views

About the definitions of direct product and direct sum of modules.

Direct product of modules can contain infinitely nonzero elements, but direct sum of modules must contain finitely nonzero elements. What's the point of the definitions? Why the definitions are ...
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0answers
21 views

Rings and modules-ideals and submodules

I am taking a course in Commutative Algebra and the following lemma in a section on localisation raised some questions. Lemma: Let M be an R-module. The following are equivalent. (1) $M=0$ (2) $M_P=...
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1answer
189 views

Why does this ring have rank $k!$?

Let $R$ be any ring free of rank $k$ over $\mathbb{Z}$ having non-zero discriminant. Let $R^{\otimes k} = R \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} R$. Then $R^{\otimes k}$ is a ring of rank ...
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0answers
21 views

$f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
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1answer
26 views

Is the tensor algebra functor a strong monoidal functor?

Let $K$ be a commutative ring. Is it true that $T(V \oplus W) \cong T(V) \otimes_K T(W)$ as $K$-algebras for any $K$-modules $V$ and $W$? The reason I ask is that I have heard it mentioned (I don't ...
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20 views

When does the unit of universal enveloping algebra adjunction fail to be injective?

The Poincaré–Birkhoff-Witt (PBW) theorem implies that if $K$ is a commutative ring and the Lie $K$-algebra $\mathfrak{g}$ is a free $K$-module, then the unit $\eta_{\mathfrak{g}}: \mathfrak{g} \to U (\...
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1answer
50 views

why the algebra $A = k[x]/(x^n)$ has finite representation type? [closed]

Suppose that $k$ is algebraically closed. Then why the algebra $A = k[x]/(x^n)$ has finite representation type? please clarify the answer.
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1answer
24 views

Question about $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$.

I am confused about the set $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$... Could someone please explain me why this corresponds to the set of pairs of symmetric $3$ by $3$ matrices? Thank you!
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1answer
63 views

What is $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p(X)} \mathbb{F}_p(\sqrt{X})$?

I am trying to understand what $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p (X)} \mathbb{F}_p(\sqrt{X})$ is. $\mathbb{F}_p(\sqrt{X})$ is the field of rational functions in $\sqrt{X}$. What is it ...
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0answers
32 views

does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
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2answers
73 views

$\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$. Does this module have a special name? Does a basis exist for every $n$? And if so, is there an ...
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2answers
41 views

$(R^{ \oplus A_1 })^{ \oplus A_2 } \cong R^{ \oplus (A_1 \times A_2)}$ - not a categorical proof

I need to prove that $(R^{ \oplus A_1 })^{ \oplus A_2 } \cong R^{ \oplus (A_1 \times A_2)}$ as $R$-modules. Recall that if $N$ is an $R$-module and $A$ is a set, then we can construct $R$-module ...
2
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1answer
34 views

Exterior algebra of a ring

In the book "Cohen-Macaulay rings" by Bruns and Herzog, the quick introduction of tensor algebra and exterior algebra left me a bit bewildered. After referring to the section on tensor algebra ...
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2answers
44 views

Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
2
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1answer
323 views

Is a group homomorphism a module homomorphism? [closed]

Let $A$, $B$ be two abelian groups, let $f$ be a homomorphism from $A$ to $B$. Now we construct two $R$-modules with a ring $R$, please show that $f$ is also a module homomorphism between them. I ...
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1answer
36 views

$R$-module $M$ which is artinian, but not noetherian [duplicate]

As I' m currently dealing with artinian and noetherian modules, I' m asking myself whether there is an artinian module which is not noetherian. I think so, but even after I thought a lot about this, ...
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1answer
35 views

Does there exist a general technique for solving systems of multivariable linear congruences

I'm aware for coprime moduli we have the CRT for solving the problem $$ \begin{matrix} a_0 x \equiv b_0 \mod m_0 \\ a_1 x \equiv b_1 \mod m_1 \\ \vdots \\ a_n x \equiv b_n \mod m_n \end{matrix} $$ ...
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0answers
51 views

Are the elements of a module also called vectors?

Are the elements of a module also called vectors? Or if someone says 'vector', are they talking only about a vector space? If no context is given, are there some standard assumptions?
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0answers
23 views

How does restriction of scalars interact with tensor products?

Say that we have a morphism of commutative rings $f: R \to S$. Does the restriction of scalars functor $f^*: S \text{Mod} \to R \text{Mod}$ commute with tensor products? In other words, I would like ...
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0answers
39 views

A monomorphism from an $R$-module to $R$

Let $R$ be a commutative ring with unity possessing an element $r$ in the singular ideal $Z(R)=$ the set of elements whose annihilators are essential in the module $R_R$, and let $M$ be a faithful $R$...
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1answer
24 views

Counit for the restriction of scalars, extension of scalars adjunction

Let $f: R \to S$ be a morphism of noncommutative rings. Let $$f_!:= S \otimes_R (-) : R \text{Mod} \to S \text{Mod}$$ denote the extension of scalars functor, and $$f^*: S \text{Mod} \to R \text{Mod}$...
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0answers
37 views

Let $M$ be any $A$-module. Is it true that $M\oplus A\cong A\oplus A$ implies that $M\cong A$? [duplicate]

Let $M$ be any $A$-module. Is it true that $M\oplus A\cong A\oplus A$ implies that $M\cong A$? I think it would be true and easy to prove with $A$ a PID, but I don't know in the general case. I ...
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1answer
45 views

Proof that a module is simple [closed]

Let $H$ be the subring of $\mathbb R^{4\times4}$ generated by $\mathbb R$ and the two matrices $$ A = \left[\begin{matrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 ...
4
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1answer
40 views

Complex representation of a quaternionic matrix

It is evident that right module $\mathbb{H}^n$ is $\mathbb{C}$-linearly isomorphic to $\mathbb{C^{2n}}$ with corresponding isomorphism $\nu : \mathbb{C^{2n}} \to\mathbb{H}^n $ given by $ \nu(a,b) ...
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2answers
253 views

Entries of the inverse of $\left[\frac{1}{x+i+j-1}\right]_{i,j\in\{1,2,\ldots,n\}}$ are polynomials in $x$.

Let $n$ be a positive integer. Define $$\textbf{A}_n(x):= \left[\frac{1}{x+i+j-1}\right]_{i,j\in\{1,2,\ldots,n\}}$$ as a matrix over the field $\mathbb{Q}(x)$ of rational functions over $\mathbb{Q}$ ...
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1answer
36 views

Why is $0_R0_M$ = $0_M$ in a left $R$-module $M$?

Given a left module $M$ over a ring $R$ where $0_M$ is the additive identity in $M$ and $0_R$ is the additive identity in $R$, I am trying to work out why $r0_M= 0_M$ for all $r \in R$. So far I have ...
0
votes
1answer
30 views

Finitely generated Hom module

Let $M, P$ be two finitely generated $A$-modules ($A$ is a commutative ring with $1$). Let also $P$ be projective. I have to prove that Hom$_A(P,M)$ is finitely generated as $A$-module. Before to ...