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

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1answer
48 views

Expressing $I/J$ in terms of quotients of the larger ring

Let $R$ be a Noetherian ring. Let $I\subseteq J$ be nonzero left ideals of $R$. Can the factor ring $I/J$ be expressed in terms of sums, quotients or submodules of rings of the form $R/K$, where $K$ ...
1
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1answer
69 views

Equivalent properties on the vanishing of Bass numbers

I'm studying on this notes. I'm finding some difficulties on proposition 12 on page 15. Let me recall what we are trying to prove: At first we are trying to prove that if inj ...
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2answers
173 views

Finitely generated $R$-module in the product of quotient field.

Let $R$ be an integral domain with the quotient field $K$. Let $M$ be a finitely generated $R$-submodule in $K^n$. Is it true that $M$ is free $R$-module?
2
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1answer
210 views

Why does this module have finite length?

Let $A$ be a noetherian local ring with maximal ideal $m$. Let $p$ be a prime ideal such that if $B=A/p$, then $\mathrm{dim}\;B=1$. Take $x\not\in p$, $x\in m$, and set $C=B/xB$. Then $C$ has finite ...
1
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1answer
59 views

$A\times B\cong M$

Let $M$ be left $R$-module and $A$ ,$B$ are two submodule of $M$ such that $A\times B\cong M$. Is there submodule $C$ such that $A+C=M$ and $A\cap C=0$? All my attempts proving the $A$ has a ...
1
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1answer
946 views

How to programmatically find the inverse of a 2x2 matrix (mod 26).

I'm trying to create a hill cipher utility. One feature I want is to be able to compute the key if you have the plaintext and ciphertext. $C$ = ciphertext matrix ($2\times 2$), $P$ = plaintext matrix ...
2
votes
2answers
220 views

Quotients of Indecomposable Modules

This is probably a really basic problem, but I am looking at the homomorphic image of an indecomposable module. Are there any standard results which help to determine when such an image will itself be ...
1
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2answers
89 views

Minimal Right Ideals in $\Bbb{C}[G]$

Let $G$ be a finite group. Consider the group algebra $\Bbb{C}[G]$ as a right module over itself. By Maschke's Theorem, $\Bbb{C}[G]$ is semisimple. How can we identify all the minimal right ideals of ...
4
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1answer
70 views

Decomposition of the injective hulls

I think this question is easy but I just cannot see how to solve it. Let $R$ be a ring and $M$ an $R$-module. Suppose $0=\bigcap_{i=1}^n N_i$ is a decomposition of $0$ with irreducible submodules of ...
1
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1answer
115 views

Primary decomposition of Noetherian submodules

There is a proof I don't get in Lang's Algebra: (page 423) Theorem 3.3. Let $M$ be a Noetherian $A$-module. Let $N$ be a submodule. Then N admits primary decomposition. Proof. We consider the ...
0
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1answer
176 views

On the injective dimension of a module

Let $R$ be a ring and $M$ an $R$-module then inj dim $M\leq i\in\mathbb{N}$ if and only if $\mathrm{Ext}^{i+1}(N,M)=0$ for every cyclic module $N$. The implication from left to right is obvious, I'm ...
2
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1answer
93 views

When the injective hull is indecomposable

Let $R$ be a ring and $M$ an $R$-module. Denote by $E(M)$ the injective hull of $M$. I was trying to prove that the following conditions are equivalent: 1) $(0)$ is meet-irreducible in $M$; 2) ...
0
votes
2answers
307 views

Direct summands of $\mathbb Z \oplus \mathbb Z$

Consider the $\mathbb Z$-module $\mathbb Z \oplus \mathbb Z$. I have seen two assertions about the direct summands of $\mathbb Z \oplus \mathbb Z$ but I have trouble with these: $(a,b)\in\mathbb Z ...
4
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1answer
155 views

Why are projective modules contained in this class of modules?

Suppose $A$ noetherian and define $G(A):=\{M: M$ is an $A$-module reflexive and Ext$^i_A(M,A)=$Ext$^i_A(M^*,A)=0$ for $i\geq1\}$ Why are projective modules contained in this class? Of course if ...
2
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1answer
67 views

Modules of a group over different fields.

Let $G$ be the cyclic group of order 2 and let $V$ be the regular $\mathbb{F}G$-module of $G$. Determine the $\mathbb{F}G$-submodules of $V$ where $\mathbb{F}$ has each of zero or positive ...
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0answers
49 views

Injectivity of maps on quotients of free but non-finite rank modules

Let $(A,\mathfrak{m})$ be a local noetherian domain. Suppose that $M$ and $N$ are free modules over $A$ and $f: M \rightarrow N$ is an $A$-module morphism. Suppose as well that $f$ is injective and ...
14
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3answers
605 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
5
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1answer
201 views

Idempotent and Hermitian vectors in Group Algebra

Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
4
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1answer
156 views

Smallest pure subgroup containing a fixed subgroup

I will ask a slightly more precise question then in the title. Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups they generate are in direct sum $\langle g_1 ...
3
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3answers
277 views

Splitting exact sequences of finite abelian groups

I would like to find a condition for an exact sequence of abelian groups $$ 0\to H\to G\to K\to 0 $$ to split. Assume for simplicity that $H=\langle h \rangle$ is cyclic, and choose a basis for $G= ...
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2answers
68 views

Endowing $A\otimes_k B$ with a $kG$-module structure

Let $k$ be a field and $G$ a group. Suppose we have $kG$-modules $A$ and $B$, and we want to consider $A\otimes_k B$ as a $kG$-module via $g(a\otimes b)=ga\otimes gb$. The module axioms are easy ...
4
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1answer
219 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
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2answers
165 views

questions about representation theory/structure theorem for finitely generated modules

I am reading a book in representation theory by James and Liebeck. They define an FG-module as: Let $V$ be a vector space over the field $F$ and let $G$ be a group. Then $V$ is an FG-module if a ...
1
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1answer
108 views

How to show flat modules stay flat after base ring extension?

If $g: A\rightarrow B$ is a ring homomorphism and $M$ is a flat $A$-module. Then $M_{B}=B\otimes_{A}M$ is flat $B$-module. We need to prove if $f:S\rightarrow T$ is injective, then $f\otimes 1: ...
2
votes
1answer
138 views

Basis for a tensor product of group algebras

Let $G$ and $H$ be groups, and $R$ a commutative ring. Then elements of $RG$ look like finite sums $\sum\limits_{g\in G}r_g\,g$, and similarly for $RH$. So $RG$ and $RH$ are $R$-modules with bases $G$ ...
3
votes
3answers
196 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
3
votes
1answer
161 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
0
votes
1answer
53 views

“Two ways” to reduce a module

Let $M$ be a module over a principal ideal domain $R$ and $\mathfrak{m}$ a maximal ideal of $R$ with residue field $R/\mathfrak{m}=k$ of characteristic $p$. Under what circumstances are the modules ...
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2answers
737 views

Universal property of free module, “converse”

Let $F$ be a free $R$-module with a basis $B$. We know that $B$ satisfies the following property: For any $R$-module $M$ and any $g:B\rightarrow M$, there exists a unique $R$-map ...
0
votes
1answer
42 views

Find $n$ value satisfying equations in $\mathbb Z_n$

Determine for which $n \in \mathbb N^*$ values the following equations in $\mathbb Z_n$ are satisfied: $[3] \cdot [5] = [3] + [5]$ $[3] \cdot [5] = [27]$ and $[3] + [5] = [11]$ I can't come up ...
3
votes
1answer
82 views

Given a group $G$ and $H\leq G$ is there a homomorphism $f: G \rightarrow G$ s.t $f(H) = id$ and $f(x) \neq id$ if $x\notin H$?

The motivation for this question is that I am trying to Read Basic Algebra 2 (despite a somewhat weak background for the book) and if this is true then it is fairly easy to show that monics in Grp are ...
3
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3answers
142 views

What is a non-degenerate module?

I know what a non-degenerate bi-linear form is, but what does it mean for say a left $R$-module $M$ to be non-degenerate? (Here $R$ is a ring without unit$) I came across a module being called ...
3
votes
3answers
122 views

Modules which are not free

I am trying to figure out the following: Given a principal ideal domain $R$ which is not a field, does there necessarily exist a module over $A$ which is not free? $A$ is a PID, so taking submodules ...
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1answer
349 views

modules is direct sum of simple submodule all of which are isomorphic to simple module

I'm looking for an article or source that deals with modules with this property: $M$ is $R$-module which is direct sum of simple submodules all of which are isomorphic to simple module. I want to ...
6
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1answer
146 views

If $I$ is injective and $P$ is projective, is $\operatorname{Hom}(P,I)$ injective?

In Bourbaki's Algebra, there is the following exercise ($A$ is an arbitrary ring with unity): Suppose that $I$ is an $(A,A)$-bimodule which is injective as a left $A$-module and a right $A$-module ...
5
votes
2answers
822 views

Universal property of tensor product [duplicate]

Possible Duplicate: Equality of two notions of tensor products over a commutative ring Let $A$ be a commutative ring. There are two common definitions of tensor product of two $A$-modules ...
4
votes
2answers
173 views

Localization and Extension of modules

Let $R$ be a commutative ring and $S$ be an $R$-algebra. Assume that $S$ is finitely generated as an $R$-module. Let $M$ and $N$ be finitely generated $S$-modules and $\mathfrak{m}$ a maximal ideal ...
8
votes
1answer
358 views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are: How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
0
votes
2answers
118 views

Direct Sum and Sum

We know if $R$ is semi simple ring and $M$ is $R$-module then for any submoldule $N$ of $M$ there exists submodule $N'$ such that $N \oplus N'=M$. Now I have trouble with sum and its supposed answer. ...
3
votes
1answer
77 views

A module with non-zero dual that does not canonically embed in its bidual

I'd like to find a ring $A$ and an $A$-module $E$ such that $E^*\neq\{0\}$ and the canonical mapping of $E$ into $E^{**}$ is not injective. As a hint (it's an exercise in Bourbaki's Algebra) I have ...
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1answer
128 views

Annihilator of a simple module 2 [duplicate]

Possible Duplicate: Annihilator of a simple module Let me ask the same question as before because I still have trouble understanding the problem. Let $R$ be a finitely generated ...
0
votes
1answer
71 views

Lifting projective indecomposable modules - Benson's vol. 1

I'm reading Benson's Representations and Cohomology I, Section 1.9. Could someone please clarify to me the following sentence at page 18 lines -7,-6,-5: "So given a simple $\bar\Lambda$-module ...
2
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0answers
256 views

Dual sequence and its exactness

I am reading Lang's Algebra and trying to fill in the gaps in my mathematical background while I train for the quals. So I came across the following exercise (chapter 20, ex. 26 in the third edition). ...
1
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1answer
95 views

Decomposition of Modules

Every finitely generated module $M$ over a principal ideal domain $R$ is isomorphic to a direct sum of cyclic modules. If $M=M_n(\mathbb Z)$, the matrices of order $n$ over the integers. What is the ...
1
vote
1answer
405 views

Annihilator of a simple module

Let $R$ be a finitely generated commutative ring and $C$ an $R$-algebra ($C$ is not necessarily commutative). Assume that $C$ is a finitely generated $R$-module. If $S$ is a simple $C$-module, then ...
3
votes
1answer
664 views

On modules over polynomial rings

Let $\mathbb{A}$ be a polynomial ring in $n$ variables over an algebraically closed field $\mathbb F$. Given a maximal ideal $\mathfrak{m}$ of $\mathbb A$, consider the quotient ...
2
votes
1answer
168 views

If $R$ is a ring s.t. $(R,+)$ is finitely generated and $P$ is a maximal ideal then $R/P$ is a finite field

Let $R$ be a commutative unitary ring and suppose that the abelian group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal of $R$. Then $R/P$ is a finite field. Well, the ...
3
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0answers
95 views

Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
2
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0answers
92 views

Projective dimension of simple module

Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $$ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $$ ...
6
votes
3answers
334 views

Finite modules over finite local rings

Let $R$ be a finite commutative local ring with identity. If $M$ is a finite $R$-module it is necessarily projective?