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

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1answer
68 views

Characterization of faithfully flat modules

This is an exercise from Rotman, introduction to homological algebra. A right $R$-module $B$ is called faithfully flat if : 1) $B$ is flat 2) If $X$ is a left $R$-module and $B \otimes_R X =0 $ ...
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1answer
185 views

Pushout of an injective map is injective

This is an exercise from Rotman , Introduction to homological algebra. Given a pushout diagram in $R$-Mod $$\begin{array} AA & \stackrel{g}{\longrightarrow} & C \\ \downarrow{f} & & ...
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1answer
149 views

Dual of Schanuel lemma

This is an exercise from Rotman, Introduction to homological algebra. Given exact sequences of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & M & \overset{i}{\longrightarrow} ...
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2answers
262 views

Quasi-hereditary algebras

Can anyone please recommend a reference (I prefer a book chapter) on Quasi-hereditary algebras? and if it is possible tell me how to prove that Schur algebras are Quasi-hereditary (or any other ...
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1answer
66 views

Applying an additive covariant functor to a sequence

Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
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1answer
64 views

$\hom_R(A,B)$ is finitely generated if $R$ is noetherian [duplicate]

This is part of an exercise I'm doing, from Rotman Introduction to homological algebra. Let $R$ be a commutative ring, and let $A$ and $B$ be finitely generated $R$-modules. Then if $R$ is ...
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1answer
55 views

Proves or counterexamples in retraction and coretractions of modules

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z} $ is not a ...
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1answer
57 views

Starting projective modules problems

Show that if $n=rs$ where $n,r,s>1$ are positive integers, then the $ \mathbb{Z}_n $-module $r \mathbb{Z}_n$ is projective but it is not free if $(r,s)=1$. Any ideas or help how to prove this ...
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1answer
91 views

Newbie into categorical proofs

Let $$ F:Mod_A \to Mod_B $$ an aditive , exact and covariant functor and $$ M ∈ Mod_A $$ and $$M_1 , M_2 $$ submodules of M . Show that $$ F( M_1\cap M_2)=F(M_1)\cap F(M_2 ) $$ $$ F( M_1+ M_2)=F(...
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2answers
129 views

Help proving a short exact sequence

Show the following sequence is an exact sequence of $\mathbb Z$-modules when $n$ is a positive integer such that $n=rs$: $$ 0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0. $$ should ...
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1answer
51 views

Proving that the following is an exact sequence

If $L$ and $N$ are submodules of $M$ show that the following is an exact sequence: $$ 0\to M/(L\cap N) \to (M/L)\oplus (M/N) \to M/(L+N )\to 0 $$ for the last morphism $g(x +L , y+N) = (x-y) + (L+N)...
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1answer
98 views

Every finitely generated module is the quotient of a finitely generated projective module.

Every finitely generated module is the quotient of a finitely generated projective module. I already find a proof of this but it uses tensor functor and flat modules propositions so im looking for a ...
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4answers
98 views

Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
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2answers
48 views

Not so usual equivalence of maximal left ideal of a ring

I was reading Foundations of Module and Ring Theory and i found this equivalence of maximal left ideal as exercise in the the first chapter: A left ideal $I$ of a ring $R$ is a maximal if and only ...
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1answer
53 views

Definition of direct summand

If M is an R-module ,and if N is a direct sum of M over an indexed set I, then is it correct to write: N= T direct sum M, where T is the direct sum of M over I-{i} , where i in I.
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2answers
137 views

If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated.

I'm answering this question: Let $M$ be a module. Show that if $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated but $A$ is finitely generated whenever $...
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1answer
131 views

Annihilator of a quotient module

Let $J$ be an ideal of $R$, and $M$ a right $R$-module. Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine $\...
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1answer
25 views

Quotient different modules by the same module to get the same module.

More precisely, If i have modules $M$ and $M'$ and $N$ and $N'$ does $M/N \cong M'/N'$ and $N \cong N'$ imply that $M \cong M'$? What if we also know that $N$ is free or that $M/N$ is free?
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0answers
90 views

If a direct sum has a projective cover, must the summands have projective covers?

In “Cover of a direct summand” it is asked to show that if a direct sum has a projective cover, and if one of the summands has a projective cover, then so does the other. I gave a solution that works ...
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1answer
23 views

Minimal Frattini Extensions

From "Construction of Finite Groups" article : "To construct minimal Frattini extensions of $H$ we have to consider all suitable irreducible $H$-modules $M$. Clearly, $M$ can be viewed as an ...
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1answer
107 views

Cover of a direct summand

Let $L,N$ be $R$-modules. If $L$ and $L \oplus N$ have projective covers, is it true that $N$ admits a projective cover?
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1answer
57 views

$\bf{Z}$-module homomorphism and $\bf{Q}$-module homorphism

$R$-modules are also $\bf{Z}$-modules and $R$-module homomorphisms are also $\bf{Z}$-module homomorphisms. If $M$ and $N$ are $\bf{Q}$-modules and $f : M \rightarrow N$ is a $\bf{Z}$-module ...
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0answers
58 views

A proposition on exact sequence of inverse limit (Lang, Algebra, p. 165)

I am trying to understand this proof. My only question is that what are the vertical maps here?
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1answer
50 views

$M_R$ is finitely generated iff Every submodule of $M_R$ is finitely generated

$M_R$ is finitely generated Every submodule of $M_R$ is finitely generated. Do the sentences above have the same meaning? Thanks for any replies.
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2answers
139 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
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1answer
94 views

Find radical of a quiver representation

How to find the radical of an acyclic quiver and without relations? what is the recipe? For example suppose $Q$ is a quiver with two vertices and two arrows from vertex $2$ to vertex $1$. Now suppose ...
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1answer
52 views

Is there a direct proof that $M_n(\mathcal k)$ is semisimple ring.

An R-module M is called semisimple if on of the following condition holds: 1) M is a direct sum of simple* submodules of M 2) M is a sum of simple submodules of M 3) For any R-submdoule N of M ...
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1answer
57 views

Some questions about the properties of submodule

I am reading a book on Abstract and Linear Algebra and saw the following theorem: "Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. i) $\cap_{t\...
2
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1answer
140 views

Exercise from Rotman, direct limit of quotient modules

Suppose $$0 \to U \to V \to V/U \to 0$$ is an exact sequence of left $R$-modules. Let $\lbrace U_i, \alpha^{i}_j\rbrace$ be a direct sequence of submodules of $U$, where $$\alpha^{i}_j : U_i \to U_j \...
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1answer
48 views

Does there always exist finitely presented submodules?

Suppose $M\neq 0$ is an $A$-module where $A$ is a commutative ring with $1$. Is it always possible to find a finitely presented submodule $N\neq 0$ of $M$? It is not interesting when the ring $A$ ...
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0answers
134 views

Length of a composition series of a module

If $A=\mathbb{C}[x,y]_{(x,y)}$, then what is the length of $A$-module $$A/(x^3-x^2y^2+y^{100},x^3-y^{999})\ ?$$ Any suggestion ?
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1answer
65 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) \...
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1answer
41 views

Extension of a group by a module

What is the definition of an extension of a group by a module? and what is the difference between extensions by groups and modules?
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3answers
173 views

Embeddings $A → B → A$, but $A \not\cong B$?

Are there any nice examples of structures (groups, modules, rings, fields) $A$ and $B$ such that there are embeddings $A → B → A$ while $A \not\cong B$? I would especially like to see an example for ...
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0answers
26 views

When is an element of this tensor product equal to $0$?

Let $A$ be an abelian group (i.e., a $\mathbb Z$-module), and consider the tensor product $\mathbb Q\otimes_{\mathbb Z} A$. I want to show that the element $\frac{1}{d}\otimes n=0$ iff there's a ...
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1answer
335 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
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2answers
175 views

Tensor product of a vector space and a field

Let $F$ be a field and $V$ a vector space of finite dimension $n$ over $F$. Let $\overline{F}$ be the algebraic closure of $F$. and let $\overline{V}=\overline{F}\otimes_F V$ the tensor product over $...
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1answer
73 views

Criterion for locally free modules of rank $1$

Let $R$ be a commutative ring and let $M$ be a finitely generated $R$-module such that the $R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is free of rank $1$ for every prime ideal $\mathfrak{p}$. Can we ...
0
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1answer
51 views

isomorphism $ \Bbb CG $ modules

Let $ \Bbb CG$ algebra, and let $ x= \sum _{g \in G} g $ . We define the module $ \Bbb Cx=\{cx: c \in C\}$. I must prove that $$ \Bbb C \cong _{ \Bbb CG} \Bbb Cx $$ Any ideas about that..thanks in ...
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1answer
172 views

How to construct a nonzero homomorphism from a module to a proper submodule?

Let $M$ be a finitely generated module over a commutative ring and $N$ be a non zero proper submodule of $M$. Then is it always possible to have a non zero homomorphism $f$ from $M$ to $N$?
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0answers
75 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
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0answers
64 views

Should a left module be an enriched functor or enriched presheaf?

In this page http://ncatlab.org/nlab/show/module#InEnrichedCategory, They defined the left module over a monoid object A as an enriched presheaf. However, consider the modules over a ring, the left ...
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1answer
55 views

What does this notation mean, which looks like direct sum?

I ran across this notation in a paper. What does $ Z^{\bigoplus n(n-1)}$ mean? I knew that $ Z^n$ or $ Z^{(n)} $ means direct sum, but never met this one.
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1answer
40 views

What does this notation mean? Perhaps related to abelian group and module.

I found this in a paper, but I don't it's meaning. Where H is an abelian group, what does $ \bigwedge ^{2} H $ mean? If H is a dual vector space, perhaps it is wedge product. But I don't know when it ...
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1answer
75 views

Algebra extension of modules question

Let $M$ be an extension of modules of $A$ by $B$ (modules), i.e. $A \leqslant M$ and $M/A \cong B$. Show that if $M$ is finitely generated then so is $B$. In the solution it just says: If $M = XR$ ...
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1answer
232 views

Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
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2answers
69 views

Quotient modules isomorphic $ \Rightarrow$ submodules isomorphic

Suppose $M$ is a module, $H$ and $K$ submodules. If $$M/H \cong M/K$$ can I conclude that $H \cong K $ ? If not, how to construct a counterexample ?
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1answer
162 views

Can one prove the existence of tensor product without explicitly constructing it? [duplicate]

R is a ring with 1. We construct tensor product $M \otimes N$ of right R-module $M$ and left R-module $N$ to basically be able to state its universal property that any R-bilinear map from $M\times N$...
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1answer
53 views

An element annihilates a module

Let $R$ be a ring, $A$ an $R$-module, $y$ belong in $R$ such that $1+y$ annihilates $A$. Then for any ideal $I$ containing $y$, prove that $IA=A$.
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1answer
37 views

Short Exact Sequences and R modules

Let $0 → A → B → C → 0$ be a short exact sequence of $R$-modules. Prove that for any $R$-module $M$ , there is a short exact sequence $0 → A \oplus M → B \oplus M → C → 0$. Can anyone please help me ...