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

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2
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0answers
17 views

Injective homomorphism from an ideal to a torsion free module over a Dedekind ring

Let $M$ be a nonzero torsion free module over a Dedekind ring $R$ and $I ⊂ R$ an ideal. Show that there is an injective $R$-module homomorphism $I → M$. My solution is to take a nonzero element ...
0
votes
1answer
13 views

Suppose $N \cong R^n $ then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by $\sum m_i \otimes e_i $

Suppose $N \cong R^n $ be free $R$ module of rank $n$ with basis $\{e_1,...,e_n\}$ and $R$ is commutative then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by ...
0
votes
0answers
12 views

The module of infinite matrices has bases with any length, isn't it? [duplicate]

Let $R$ be a ring. We consider matrices of elements from $R$ with the following properties: The sizes of a matrix is infinite; Any row of a matrix have a finite number of nonzero elements of $R$ ...
0
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0answers
19 views

What is the rank of $\Bbb{Z}_m\oplus\Bbb{Z}_n$?

At first glance, the rank seems to be $2$. The basis elements seem to be $(0,1), (1,0)$. However, how is scalar multiplication defined? Is $a(p,q)=(ap,q)=(p,aq)$? So if the module under ...
0
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0answers
29 views

Proving “up to unique isomorphism”: Universal property of cokernel

This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following: Suppose you have homomorphism $f: M ...
1
vote
1answer
55 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from ...
1
vote
1answer
32 views

Submodules finitely generated if sum and intersection are finitely generated? [duplicate]

Let $M,N$ be submodules of an R-module $L$, where $R$ is a commutative ring with unity. I would like to prove that if $M+N$ and $M \cap N$ are finitely generated so are $M$ and $N$. Trying to combine ...
2
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0answers
31 views

Any characterization of rings over which submodules of left free modules are free?

Let $R$ be a ring (with identity, but not necessarily commutative). If given the presumption that for any left free $R$-module $M$, every submodule of $M$ is free, what can we say about $R$? ...
1
vote
1answer
60 views

Extending a given element of a free abelian group to a basis

Let $V$ be a free $\mathbb{Z}$-module (or a free abelian group) with basis $(v_1$, $v_2$, $v_3)$. Denote by $x$ an arbitrary nonzero element of $V$, say $x=mv_1+nv_2+kv_3 \in V \setminus \{0\}$; here ...
0
votes
2answers
40 views

Why does dual basis not span the dual space when the given vector space V is infinite dimensional?

We have a vector space V with basis $\mathcal{A}$. I have to show that the set $\mathcal{A}$* = { v* | v$\in\mathcal{A}$} does not span the dual space V*. I can not see why dual basis fails to span ...
2
votes
1answer
48 views

Torsion-free divisible module over a commutative integral domain is injective. [duplicate]

This question is from the book Basic Algebra by P.M. Cohn. Show that a torsion free divisible module over a commutative integral domain is injective.
1
vote
1answer
36 views

Lifting homomorphism when module is direct summand of free module

Let $N,M,M',N'$ be modules such that $F = N \oplus N'$ is a free module. Let further $f: M \rightarrow M''$ be a surjective homomorphism. I would like to show that for any homomorphism $\phi: N ...
1
vote
1answer
22 views

Question about proof of Krull principal ideal theorem

How can we explain the following step in the proof of Krull principal ideal theorem: $l\{ ((z):x^n)/(z) \}$ or $l\{ ((x^n):z)/(x^n) \}$ is finite? $l(M)$ - length of module.
2
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0answers
69 views

Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an ...
1
vote
1answer
27 views

Applying hom-functor on short exact sequence

Let $N, M, M', M''$ be $R$-modules. Given homomorphisms $f: M' \rightarrow M$ $g: M \rightarrow M''$ $\psi: N \rightarrow M$ with $\operatorname{im} f = \ker g$, $g \circ \psi = 0$ and $g$ ...
1
vote
1answer
49 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 ...
1
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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 ...
-1
votes
1answer
55 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 ...
0
votes
1answer
39 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 ...
0
votes
0answers
9 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 $N$?
0
votes
1answer
33 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 ...
0
votes
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 ...
0
votes
2answers
39 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 ...
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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.
-1
votes
1answer
36 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. ...
0
votes
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 ...
0
votes
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, ...
2
votes
1answer
37 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 ...
0
votes
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
votes
0answers
209 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 ...
0
votes
1answer
43 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, ...
0
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1answer
25 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 = ...
0
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0answers
11 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 ...
0
votes
1answer
23 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
34 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 ...
4
votes
1answer
140 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
votes
1answer
49 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 $m\in M$. ...
2
votes
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 ...
0
votes
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? ...
1
vote
1answer
21 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 $V_{\alpha}$ ...
0
votes
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, ...
0
votes
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. ...
1
vote
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 ...
4
votes
2answers
35 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
votes
0answers
40 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 ...
0
votes
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 ...
-1
votes
1answer
60 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. ...