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

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Wedderburn rings, Projective Modules

Prove that an artinian ring with no nonzero nilpotent elements and no non trivial central idempotents is a division ring.
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For every ring R, every left R-module M can be imbedded as a submodule of an injective left R-module [closed]

Prove that for every ring $R$, every left $R$-module $M$ can be imbedded as a submodule of an injective left $R$-module
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Maximal $\mathbb Z_p$-orders in $\mathbb Q_p[G]$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. ...
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23 views

Use the prime factorization of $\phi(321)$ to determine possible orders of units mod 321

Use the prime factorization of $\phi(321)$ to determine possible orders of units mod 321. (Your list should have fewer than ten numbers in it). My attempt: $\phi(321)$ = $\phi(3)\phi(107)$ = $2(3)$ ...
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0answers
26 views

Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any ...
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2answers
26 views

Generating system for $R$-module $M$ still generating $\mathfrak aM$?

Let $M$ be a module over a commutative ring $R$ with unity and $\mathfrak a$ an ideal in $R$. Let $m_1, ..., m_n \in M$ be a generating system for $M$ and $z \in\mathfrak a M$. Then there are $a_1, ...
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1answer
33 views

Module over an infinite dimensional algebra

I have two question related to infinite dimensional algebra I have been seen a lot of example about Module over a finite dimensional $k-$algebra, but I could not find a literature about Module ...
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2answers
26 views

Dimension of a basis for a module using the quotient map

I'm going through a proof that the size of a basis for a module is uniquely determined and can't see why the following line from my lecture notes follows: "We now claim that if $X$ is a basis for ...
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206 views

Computing (the ring structure of) $\mathrm{Ext}^\bullet_R(k,k)$ for $R=k[x]/(x^2)$

Let $k$ be some field (say of characteristic zero, if it matters) and define $$R=k[x]/(x^2).$$ I want to compute $$\mathrm{Ext}^\bullet_R(k,k)$$ and, in particular, the ring structure on it (though I ...
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1answer
29 views

What is the cokernel of the map $A:\mathbb Z^2\to \mathbb Z^7$?

Consider the free $\mathbb Z$ - modules $\mathbb Z^2$ and $\mathbb Z^7$ and consider the $\mathbb Z$ - module map $A:\mathbb Z^2\to \mathbb Z^7$ given by, ...
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1answer
23 views

Support of a module over polynomials: intuitive meaning

I find myself working with modules over complex polynomials in more variables, say $C[u_1,\dots,u_l]$. One can introduce the concept of support of the module $H$ as \begin{equation} supp H = ...
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9 views

Equivalence between absolutely pure modules and FP-injective modules

Let $R$ b a ring and $Q_{R}$ a right $R$-module. Show that the following conditions are equivalent for $Q_{R}$ : $(i)$ $Q_{R}$ is FP-injective, that is, given any short exact sequence $0 \rightarrow ...
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1answer
22 views

Find a vector $v$ such that $V=\mathbb{C}[T]\cdot v.$

Let $V=\mathbb{C}^2$ and let $\alpha:V\to V$ be the $\mathbb{C}$-linear map given by \begin{equation*} \begin{pmatrix} a\\ b \end{pmatrix} \longmapsto \begin{pmatrix} 2 & 3\\ -3 & 8 ...
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0answers
32 views

Reflexive Graded Module

Let $R=k[x_1,\dots,x_d]$ be a polynomial ring and $M=M_0\oplus M_1\oplus M_2\oplus\cdots$ be a graded $R$-module. Is it true that $M$ is reflexive as an $R$-module if and only if $M_i$ is reflexive as ...
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1answer
136 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
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1answer
36 views

Support and Annihilator of Tensor Product of Modules

Let $M$ and $N$ be $R$-modules. Let $\mathrm{Supp}(M)$ be the set of primes $P$ such that $M_P\neq 0$, and let $\mathrm{Ann}(M)$ be the ideal of elements $r\in R$ such that $rm=0$ for all $m\in M$. ...
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1answer
38 views

Hartshorne Deformation theory Exercise7.1

This is a exercise in "Deformation Theory [Hartshorne]". Let C be a local Artin ring with residue field k. Let X be a scheme flat over C , and let$X_0=X\times _Ck$. If F is a coherent sheaf on X ...
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18 views

An ideal of $\mathcal O_K$ [duplicate]

Let $K$ be a number field and let $\mathcal O_K$ be its ring of integers. Let $I$ be a non-zero ideal in $\mathcal O_K$. If $I$ is free as a $\mathcal O_K$-module then is it a principal ideal? ...
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1answer
20 views

Vector spaces: understanding a step towards canonical forms

I am studying canonical forms of matrices, and I'm a little stuck on something: Consider a finite dimensional $\mathbb{F}-$vector space $V$ and an endomorphism $\alpha:\;V\to V$, and let $V_{\alpha}$ ...
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1answer
43 views

Is $\mathrm{Hom}_{R}(A,\prod_{i\in I} B_{i}) \cong \prod_{i\in I}\mathrm{Hom}_{R}(A, B_{i})$?

Let $R$ be a ring, $A$ is a left $R$-module, and $(B_{i})_{i\in I}$ is a family of left $R$-module. We have: $$\mathrm{Hom}_{R}(A,\prod_{i\in I} B_{i}) \cong \prod_{i\in I}\mathrm{Hom}_{R}(A, ...
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1answer
20 views

two sided ideals and idempontents

Let I be a two sided ideal of R. Prove that I=eR for some central idempotent e ϵ R if and only if R=I+J for some two sided ideal J. When this occurs, show that e and J are uniquely determined by I. ...
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1answer
23 views

Homomorphism of Matrices over Extended Field

Let $E$ and $F$ be fields such that $E$ is an extension field of $F$ and $[E:F] = n$. Let $M_k(F)$ denote the ring of $k \times k$ matrices over $F$. Does there exists a homomorphism from $M_k(E)$ to ...
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32 views

Multiplication map for algebras in the sense of Hopf algebra

Let $R$ be a commutative ring with unity and $A$ a ring with unity. In the definition of algebras, we require a multiplication map $\mu: A\otimes A \rightarrow A$ satisfying certain property. Indeed, ...
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36 views

A question on Auslander-Bridger transpose

I am learning Auslander-Reiten Theory. When I read the book Frobenius Algebras I. Basic Representation Theory, I have some problems. On page 236-237, The Proposition Proposition 4.5. Let $M$ and ...
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17 views

Let $R$ be a ring, considered as an $R-$module. How to determine all module homomorphisms? [duplicate]

Let $R$ be a ring, considered as an $R-$module. How can I determine all module homomorphisms from $M\rightarrow M$, and $R\rightarrow R$? (M is just a random module)
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1answer
28 views

The set of homomorphisms of modules is a module

Suppose $M,N$ are $R$-modules. Let $Hom_R(M,N)$ be the set of all $R$-module homomorphisms from $M$ to $N$ with operations defined by $(\phi+\psi)(m)=\phi(m)+\psi(m)$ and $(r\phi)(m)=r(\phi(m))$. How ...
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58 views

Square Root of $5$ mod $10^{9}+7$ [closed]

$My$ $Current$ $Knowledge:$ We can find it if 5 is a $Quadratic$ $residue$ modulo p and where p is prime and we can check it using $Euler$ $criterion$. I cannot able to find the root(5)mod 1000000007. ...
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Equivalent conditions for a short pure exact sequence

Let $L$, $N$, and $M$ be right $R$-modules and let $\widehat{L}=\mathrm{Hom}_{\mathbb{Z}}\left(L,\mathbb{Q}/\mathbb{Z} \right)$ ($\widehat{N}$ and $\widehat{M}$ are defined analogously). Show that ...
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1answer
17 views

Cardinality of “linearly independent” sets in a free module.

Let $R$ be a commutative ring with identity, and let $R^{N}$ be the free $R$-module of dimension $N$. A set $\{e_{1},\ldots, e_{m}\}\subseteq R^{N}$ is linearly dependent if there exist ...
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1answer
31 views

Projective Dimension and Schanuel's Lemma

Let $R$ be a ring and $M$ a (say, left) $R$-module of projective dimension $n$. According to Noncommutative Noetherian Rings, any projective resolution of $M$ can be terminated at length $n$, and this ...
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Why is the minimal polynomial of linear transformation $T$ the lcm of all elementary divisor

Suppose you have a vector space $V$ over $K$ and you give $V$ a $K[x]$ module structure via a linear operator $T$. Why then is the minimal polynomial of $T$ necessarily the lowest common multiple of ...
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15 views

Equivalence of definitions of minimal polynomial of a linear transformation $T$ over a vector space

Let $V$ be a vector space over $K$ and let $T\in End_{K}(V)$. Then $V$ has a $K[x]$-module structure via the operation defined as $f(x)v=f(T(v))$. Using this we can know a lot about the structure of ...
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22 views

Exterior power of a free module [duplicate]

Suppose $M$ is a free R module where $R$ is commutative and unital. Is $\Lambda^n M$, the nth exterior power of $M$ free?
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1answer
32 views

Additivity of direct limits

I am trying to understand the proof of a Lemma from Greenberg's Algebraic Topology. It says: Let $I$ be a directed set and $M_i$ a family of modules. Suppose that for every $i\in I$, we have ...
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1answer
38 views

Isomorphism between sum of submodules and their direct sum

Let $A$ be a commutative ring, let $M$ be an $A$-module, and let $M'$ and $M''$ be submodules of $M$ such that $M' \cap M'' = \{0\}$. Prove that $M' + M'' \cong M' \bigoplus M''$. I believe ...
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1answer
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If $P/I_kP$ are finitely-generated, is it true that $P/IP$ is finitely-generated where $I=\bigcap I_k$?

I'm looking into some old results on "big projectives'', and trying to understand some steps. Assume that $R$ is a (commutative) ring and $I_1,\ldots,I_n$ are ideals. Let $I=I_1\cap\cdots\cap I_n$ be ...
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1answer
45 views

Image is zero in every module homorphism

Suppose $R$ is a commutative ring, $A$ an $R$-module, and $A'$ is an $R$-submodule of $A$ such that for every $f \in \mathrm {Hom}_R(A,A')$ we have $f(A)=0$. Will this imply $A'=0$? I am not ...
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1answer
34 views

Can somebody explain me this proof on Kasch book (Modules and rings)?

I have a question about one step in the proof of Proposition 13.2.6: If $R_R$ is injective and ${_R}R$ is noetherian then $R_R$ is a cogenerator and $R$ is artinian on both sides. In the proof of ...
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36 views

Prove $\dim(V) \leq 2$

Consider $V$ a vector space over a field $K$. Consider a linear map $T:V\to V$ such that $T^2 = T$. Suppose $V$ is a cyclic $K[x]$-module. Prove that $\dim(V) \leq 2$. I know that the vector ...
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1answer
48 views

Ascending chain - Isomorphism theorems

So that what I did so far : Interesting results : $f$ is injective $\iff$ $\ker f = \{0\}$ $\ker f$ is a submodule of $X$ As $X$ is noetherian, then each ascending chain $M_1 \subset M_2 \subset ...
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1answer
34 views

Free Modules of Rank n

I'm prepping for prelim exams and realized I have a fundamental gap in understanding of free modules. I'm trying to answer the following question: Let $M = \mathbb{Z}^n$ and denote by $pM = ...
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1answer
52 views

Modules over ring of polynomials in several indeterminates.

I read this proposition and its proof in the Dummit's book (It's not exactly like this, but this is the idea of the proposition). Let $F$ be a field, $V$ a vector space over $F[x]$ and $T\in ...
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1answer
26 views

The order ideal of a finitely generated module over a Dedekind domain

Let $R$ be a Dedekind domain and $M$ be a nonzero finitely generated torsion $R$-module. In Curtis and Reiner's Methods of Representation Theory it states that $M$ has a composition series and if its ...
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2answers
36 views

Tensor algebra with semidirect product

I am having trouble parsing the following problem: Let $F$ be a field, $A$ an $F$-algebra with 1 (not necessarily commutative) and let $M$ be an $F$-vector space which is also an $A$-bimodule. Let ...
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78 views

Is it true that $\operatorname{Hom}_R(M,N) \otimes_R P=\operatorname{Hom}_R(M ,N \otimes P)$?

Let $M, N, P$ be modules over a commutative ring $R$. The above identity is true for $R$ a field: Since the RHS $\cong$ (by passing to double dual) $$\operatorname{Hom}(N^* \otimes P^*, ...
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45 views

Idempotent endomorphisms generate a direct sum decomposition

Show that there is a one-to-one correspondence between indempotents $e\in\operatorname{End}_R(V)$ and direct sum decomposition $V=X\dotplus Y$. Attempt: We have $e^2 = e$. Therefore, we can write an ...
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1answer
44 views

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so.

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so. In between step is $I^j/I^{j+1}$ is noetherian (artinian) $\forall j$. I ...
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1answer
35 views

Example of a non noetherian module $M$ s.t. $M/IM$ is noetherian

What is an example of a non noetherian module $M$ s.t. $M/IM$ is noetherian? What is an example of a non artinian module $M$ s.t. $M/IM$ is artinian? What I was thinking that $k[x_1,...,x_n,...]$ ...
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1answer
35 views

For which $n$ is $\mathbb{Z}_2$ a $\mathbb{Z}_n$-module?

Determine the values ​​of $n$ for which $\mathbb{Z}_2$ is a $\mathbb{Z}_n$-module (with action given by "the product $\mathbb{Z}_2$"). If $n=2k+1,$ $k\geq 1$, then: \begin{align*} ...
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1answer
25 views

Confusion regarding finding invariant factors of a matrix.

So I'm having a bit of trouble determining invariant factors of a matrix. Say we have $$ \begin{bmatrix} 2 &0 &0 \\ 0 &9 &0 \\ 0 &0 &6 \end{bmatrix} $$ and I want to find the ...