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

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1answer
52 views

Natural action of $\mathbb{Z}G$ on $\mathbb{Z}$?

I'm studying projective modules and I'm having problem coming up with (or understanding) examples of non-free projective modules. I got that when a ring is a direct sum $R = A \oplus B$, both $A$ and $...
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0answers
34 views

Understanding isomorphism of Hom's

During some reading I came across the statement $$Hom_R(M,N) \otimes R/I \cong Hom_{R/I}(M/IM,N/IN) $$ where $M,N$ are $R$-modules and $I$ an ideal of the commutative ring $R$. This is proved, under ...
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1answer
59 views

What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...
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1answer
41 views

Commutativity of ring $R$ necessary for $\mathrm{Hom}_R(M,M')$ being an $R$-module

Why do we need $R$ to be commutative if we want $\mathrm{Hom}_R(M,M')$ (where $M$ and $M'$ are $R$-modules) to be an $R$-module itself? I tried to find out which axiom for modules does not hold if $...
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0answers
10 views

Submodules of direct sum of simple modules [duplicate]

Is it true that if $N$ is a simple module, then the only non trivial submodule of $N\oplus N$ is $N$? I am aware of the "diagonal" submodule $\{(x,x)\mid x\in N\}$, but that is isomorphic to $N$?
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1answer
35 views

Abelian group structure and structure of Z[i]

Let $F = \Bbb{Z_p}$. For which prime integers $p$ does the additive group $F^1$ have a structure of $\Bbb{Z}[i]$-module? How about for $F^2$? I'm not really sure on how to approach this question, do ...
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0answers
74 views

Infinitely generated torsion free modules over PID

Let $R$ be a PID and $\mathbf{V}$ a torsion-free $R$-module, not necessarily finitely generated. If I understand it correctly, every rank 1 submodule of $\mathbf{V}$ is isomorphic to a submodule of $Q(...
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1answer
38 views

Showing that $A^n/M^n \cong (A/M)^n$

I am just trying to show that for $M$ a maximal ideal we have an isomorphism of $A$-modules $A^n / M^n \cong (A / M)^n$. I wanted to make sure that I have correctly understood the concepts and what ...
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2answers
40 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated $A$-...
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0answers
8 views

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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1answer
40 views

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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0answers
16 views

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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1answer
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
27 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 $L$-...
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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
41 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
29 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 ...
3
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0answers
212 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
51 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, $$\left(\begin{array}*&1&1&0&-...
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1answer
27 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 = \cap_{f:...
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0answers
12 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
24 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 \end{...
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0answers
35 views

projective resolution for an $I$-torsion $R$-module

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $M$ be an $I$-torsion $R$-module. We know that there exists an injective resolution of $M$ in which each ...
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0answers
36 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
148 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 ...
0
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1answer
61 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
42 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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0answers
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
22 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: $$\mathrm{Hom}_{R}(A,\prod_{i\in I} B_{i}) \cong \prod_{i\in I}\mathrm{Hom}_{R}(A, B_{i}).$$...
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1answer
21 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
24 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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2answers
37 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, ...
2
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1answer
58 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, there is the following Proposition 4.5. Let $...
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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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1answer
61 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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0answers
25 views

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 the ...
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1answer
20 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 $\alpha_{1},\...
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1answer
41 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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0answers
16 views

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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0answers
16 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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0answers
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
37 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 $M_i=N_i\...
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1answer
39 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 I ...
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1answer
14 views

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
48 views

Image is zero in every module homorphism [closed]

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 sure ...
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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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1answer
37 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 space $...
2
votes
1answer
50 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 \...
2
votes
1answer
38 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 = p\mathbb{...