-1
votes
0answers
27 views

ring and module problem

Let $$F=\mathbb{R}$$ $$V=\mathbb{R}^{4}$$ consider two matrices $$S1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 ...
1
vote
0answers
51 views

Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
1
vote
1answer
26 views

Free modules and ideals

I am trying to show that an ideal I of R=$\mathbb{C}[x_1,x_2]$ generated by $x_1, x_2$ is free R-module. I am trying to show that I has a basis of the two generators given above. But I am not able to ...
1
vote
0answers
67 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
2
votes
1answer
33 views

What is the difference between a module of finite rank and finitely generated module.

R is an integral domain and every module we talk about is an R-module. If a module is finitely generated then obviously every element of the module can be written as finite R-linear combination of the ...
1
vote
1answer
19 views

Show polynomials $I$ is not finitely generated as $R$-module

Let $R=\{a_0+a_1X+\cdots+a_nX^n\;|\;a_0\in\mathbb{Z},a_1,a_2,\cdots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}_{\geq 0}\}$ and $I=\{a_1X+\cdots+a_nX^n\;|\;a_1,a_2,\cdots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}^+\}$. ...
0
votes
0answers
31 views

Question about singular homology

in order to prove that $H_0(X)\simeq \mathbb{F}$, $\mathbb{F}$ is the unitary commutative ring we have to prove that $C_0(X)/B_0(X)\simeq \mathbb{F}$ since we have that $C_0(X)$ is generated by the ...
0
votes
0answers
39 views

Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
0
votes
0answers
25 views

Linearly Independent and Span Proof

Let R be a field, M be an R-module, $X \subseteq M$. Show that $X$ is linearly independent if and only if $x$ not $\in$ span (X\ {x}) for each $x \in X$ (I'm not sure how to write the symbol for ...
0
votes
1answer
26 views

Span and Smallest Submodule Proof

Let R be a ring, M a R-module, and $X \subseteq M$ Show that span$(X)$ is the smallest submodule of R containing X. My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and ...
0
votes
2answers
34 views

Cardinalities of bases of a free $R$ module are same? [duplicate]

Let $R$ be a ring with no zero divisiors such that for all $r,s\in R$ there exist $a,b\in R$ not both zero with $ar+bs=0$. If $R=K\oplus L$ then $K=0$ or $L=0$. if $R$ has an idendity then any two ...
3
votes
0answers
33 views

On the Bass numbers of a local ring

Assume $R=k[x,y]/(x^2,xy,y^2)$, I would like to calculate the dimension as $k$-vectorspace of $\mathrm{Ext}^i_R(k,R)$. I see that as vector-space $\mathrm{Ext}^i_R(k,k)\cong k^{2^{i+1}}$, is it true ...
1
vote
1answer
23 views

Hom($P$, $R$) $\neq 0 $ if $P$ is a nonzero projective left $R$-module (Rotman)

I've found this exercise, number $3.11$ from Introduction to homological algebra. Prove that $\operatorname{Hom}(P, R) \neq 0 $ if $P$ is a nonzero projective left $R$-module. Any hint?
3
votes
1answer
21 views

Example of a ring whose left modules are all free but has some non-free right modules

As the question says, I'm looking for a ring with all free left modules but some non-free right modules. I had thought about looking for a ring not isomorphic to its opposite and try and use that a ...
1
vote
1answer
51 views

why $RM \neq 0$

Definition: we said that $M$ is simple if $RM \neq 0$ and $M$ has no proper submodule. i couldnt understand why it must be $RM \neq 0$ could you please explain why ? Thank you for your helping.
0
votes
0answers
16 views

prove that if A is a finitely generated faithfully right $\Bbb R$-module

prove that if A is a finitely generated faithfully right $\Bbb R$-module ,then there exists $\Bbb { B\le A_R}$ such that $\Bbb {A/B} $ is faithful but $\Bbb {A/C}$ is unfaithful for all $\Bbb { ...
-2
votes
1answer
66 views

Surjective Implies Injective for R-Homomorphism on Finitely Generated Module [duplicate]

Let $M$ be a finitely generated module over a ring $R$, and let $f$ be an $R$-homomorphism from $M$ to itself. Does $f$ injective imply $f$ surjective? Does $f$ surjective imply $f$ injective? I have ...
9
votes
3answers
72 views

An example of a noncommutative PID

It's well known that when a ring $R$ is a PID, every submodule of a free $R$-module is free. I'm interested in cases when the converse holds -- that is, in rings $R$ which have the property that every ...
3
votes
1answer
62 views

Structure Theorem For PIDs

So, I'm a biologist at KCL, but I quite like mathematics and so am going through a book of exercises in algebra. Unfortunately, I've run into a problem in trying to answer some of the questions. I've ...
1
vote
0answers
17 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
1
vote
0answers
40 views

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
0
votes
0answers
11 views

A Ring with all Cyclic R-modules has Finite Projective Dimension but Infinite Global Dimension.

Can anybody give me an example of a ring $R$ with the property that each cyclic $R$-module has finite projective dimension even though the Global Dimension of $R$ is equal in infinity?
0
votes
0answers
16 views

Exact Sequences of R-Modules

Here's a lemma in A Course in Ring Theory by Passman. In the proof it is mentioned, "But the kernel of the combined epimorphism $P\rightarrow B\rightarrow C$ is clearly equal to $E$". I don't ...
1
vote
1answer
15 views

Projective Dimension and Supremum

Here is a lemma that appears in A Course in Ring Theory by Passman. In the last section of the proof the writer shows that, $\mbox{pd }A_i\leq n\iff \mbox{pd }A\leq n$ and finishes the proof. I don't ...
2
votes
1answer
82 views

Is always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?

Let $A$ be a commutative ring. Let $f \in A$. Let $A_f= A\left [ \frac{1}{f}\right ]$. Let $\hat{A}$ the $f$-adic completion of $A$. Is it always (even when $A$ is not noetherian) true that $$\hat{A} ...
2
votes
2answers
61 views

Question concerning $M_1\cap K=M_2\cap K$ and $M_1+K=M_2+K$

I have a ring $R$ with $K\le M$ and submodules $M_1,M_2$. If we have that: $$M_1\cap K=M_2\cap K \text{ and } M_1+K=M_2+K$$ can we conclude that $M_1=M_2$? I don't think that this is true ...
0
votes
1answer
26 views

Is quotient module finitely generated?

Suppose $R$ be any ring containing left ideal $I$. Then $I$ is submodule of $R$, so $R/I$ is R-module. My question is, is $R/I$ always a finitely generated?
1
vote
1answer
42 views

Simple $R$-module

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true: 1) $N$ has a finite number of submodules. 2) $\operatorname{Hom}_R(N,N)$ is a division ring. 3) ...
0
votes
1answer
37 views

associated $\mathbb{C}[t]$- module is cyclic iff cyclic vector exists

I'm stuck on a part of a question: if $T : V \rightarrow V$ is a linear endomorphism of a $\mathbb{C}$-vector space $V$, then the associated $\mathbb{C}[t]$- module is cyclic (that is $V ...
0
votes
0answers
21 views

Extension of Scalars as Equivalence Classes

Given a ring morphism $f\colon R\to S$, there is a functor $f_!\colon RMod\to SMod$, $N\mapsto S\otimes_R N$ called "the extension of scalars". My confusion arises from a statement in nLab's article ...
0
votes
1answer
23 views

modules finite congenerated are closed under extensions

I have to prove some properties about modules finite cogenerated, I´ve already prove that mmodules finite cogenerated are closed under submodules, finite direct sums, but I can´t see how to prove that ...
1
vote
1answer
31 views

The sum of all right ideals isomorphic as modules to a simple module is an ideal

I could use some help on the following problem. Let R be a ring. (a) If $r \in R$ and $U$ is a minimal right ideal of $R$, show that either $rU=0$, or that $rU$ and $U$ are isomorphic right ...
0
votes
1answer
72 views

$m$-primary ideal and $M\otimes_{A} A/m \neq 0$

Let $A$ be a commutative local ring with maximal ideal $m$. Let $M$ be a (not necessarily finitely generated) $A$-module. Let $x_{1},\dots,x_{n}$ be an $M$-regular sequence such that ...
0
votes
1answer
42 views

If M is free with a finite basis then every basis of M over R is finite and has the same number of elements.

Stuck on a proof in my lecture notes. Proposition: Let $R$ be a commutative ring and let $M$ be an $R$-module. If $M$ is free with a finite basis then every basis of $M$ over $R$ is finite and has ...
2
votes
2answers
14 views

Quotienting a direct sum by one of its factors

If we have a direct sum of $R$-modules. say $M_1\oplus M_2$ does it then follow that $(M_1\oplus M_2)/M_1\cong M_2$ This seems like it should be the case but I can't think of a way to prove ...
2
votes
1answer
123 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
3
votes
1answer
73 views

Showing $M\cong M'\oplus M''$ given an exact sequence

I am struggling with the following question: $R$ is a ring. $$M'\overset{f}{\longrightarrow} M\overset{g}{\longrightarrow} M''$$ are homomorphisms of $R$-modules such that for any $R$-module $N$, the ...
0
votes
1answer
41 views

Coproducts and direct products

Is the arbitrary direct sum of modules a submodule of their coproduct? Ie is $\underset{i \in I}{\coprod} M_i \cong \underset{i \in I}{\bigoplus} M_i$... if not then if each $M_i$ where to be ...
1
vote
1answer
28 views

direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
4
votes
0answers
63 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace, I would ...
0
votes
1answer
77 views

Relation between Jacobson radical and composition series

Let $R$ be a not necessarily commutative ring with 1. Suppose $R$, viewed as a right $R$-module, has a finite composition series with non-isomorphic composition factors. Prove that the Jacobson ...
6
votes
2answers
95 views

If $k\subset R\subset k[x]$, then $R$ is Noetherian?

Is there a way to prove that any subring $R$ of the polynomial over a field $k$ such that $k\subset R$ is Notherian without appealing to integral extensions, Eakin-Nagata, etc.? The reason I ask is ...
1
vote
0answers
41 views

Enveloping Algebra equal to algebra

Let $R$ be a unital associative ring, $A$ be an associative $R$-algebra of finite dimension, and $A^e$ its enveloping algebra. What are the requirements on $A$, so that $A^e \cong A$ (as ...
1
vote
1answer
36 views

Epimorphism that is not monomorphism from $M\rightarrow M$

I have just finished an exercise where I prove that if $M$ is a module with acc then any epimorphism $f:M\rightarrow M$ but be an isomorphism. I then had a think about examples of non-noetherian ...
3
votes
2answers
43 views

Is it possible that a scalar times a submodule is not a submodule?

Does there exist a ring $A$ and an $A$-module $X,$ such that for some $a \in A$ and some submodule $Y \subseteq X$, it holds that $aY$ is not a submodule? If $A$ is commutative, this is clearly ...
2
votes
1answer
133 views

Polynomial is zero for induced mapping of rings

Let $R$ be a commutative ring, and $M$ a finitely-generated free $R$-module. Let $\phi:M\rightarrow M$ be an $R$-linear map, and $P_\phi(X)$ the characteristic polynomial of $\phi$. Let ...
2
votes
1answer
48 views

Matrix algebras

Let $k$ be any field, then we know that every finite dimensional semi simple algebra $A$ is isomorphic to a direct product of matrix algebras with entries over a division ring. Assume that we require ...
0
votes
0answers
41 views

Modules,rings and definitions

Is there a source available with (almost) all definitions from ring and module theory,all in ONE place without theorems.There are books on module theory where are freely used unusual notions like ...
1
vote
1answer
65 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
6
votes
0answers
137 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...