1
vote
1answer
28 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
2
votes
2answers
22 views

A direct limit concerning some homomorphisms

In an algebra text there is the following argument I am stuck in the last part of which: "Let $f:B→C$ be an epimorphism in the category of $R$-modules, and $D=∑_{n=1}^∞c_nR$ be a countably generated ...
1
vote
1answer
33 views

Existence of a maximal submodule

Let $R$ be a left Artinian ring and let $M$ be a nonzero left $R$-module. Prove that $M$ has at least one maximal submodule. I don't really know where to start. Any hint? Thanks!
1
vote
0answers
25 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
2
votes
3answers
61 views

Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...
2
votes
3answers
49 views

Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
2
votes
1answer
42 views

Show that $D \cong {\rm End}_A(D^n)$ where $D$ is a division algebra and $A\cong M_n(D)$

Define $A$ to be a finite dimensional simple algebra over a field $k$. $D$ is a $k$-division algebra (not necessarily commutative) such that $A\cong M_n(D)$ for some integer $n$. Let $L$ be a minimal ...
1
vote
1answer
52 views

A Direct Sum of Members of a Certain Class of Modules

Let $S$ be a class of $R$-modules and let an $R$-module $M$ be countably generated. Suppose that, for every direct summand $K$ of $M$, each element of $K$ belongs to a direct summand of $K$ that is ...
1
vote
1answer
51 views

A condition of equivalence of flatness and projectiveness

This is a problem in "Foundations of Module and Ring Theory" of Wisbauer: " Let $R$ be a subring of the ring $S$ containing the unit of $S$. Show that a flat $R$-module $N$ is projective if and only ...
1
vote
0answers
42 views

Simple Cogenerator for the category of left $R$-modules

I want to prove that if the category of left $R$-modules has a simple cogenerator $M$, then $R$ is a simple ring. (Here, $R$ is an arbitrary ring with $1$.) My try is to take an ideal $I$ of $R$ ...
2
votes
0answers
70 views

Direct product of direct sum of a flat module

I have a problem concerning flat modules: Let $M$ be an $R$-module such that the direct product $M^A$ is flat for all sets $A$. I want to prove that $(M^{(B)})^A$ is also flat for any sets $A$ ...
5
votes
0answers
54 views

When is an $R$-projective module a projective module?

Let $R$ be a semiperfect ring. Is it a true fact that every $R$-projective module $M$ with $Rad(M)$ superfluous in $M$ is projective? I could not reach a good result using just the fact that ...
5
votes
1answer
69 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
1
vote
0answers
29 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
0
votes
0answers
41 views

Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
1
vote
1answer
43 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 ...
1
vote
1answer
25 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 ...
0
votes
1answer
29 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 ...
1
vote
1answer
55 views

A statement equivalent to flatness

If $R$ is a ring with identity and $P$ is a flat right $R$-module, it is a fact that any $R$-homomorphism $f$ from a finitely presented right $R$-module $M$ to $P$ factors through a finitely generated ...
2
votes
1answer
81 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
1
vote
0answers
48 views

Tensor product of two simple modules

Let $k$ be a field of arbitrary characteristic. Suppose that $A$ and $B$ are finite dimensional $k$-algebras. Let $S$ be a finite dimensional simple $A$-module and let $T$ be a finite dimensional ...
0
votes
1answer
38 views

How to show Not a Free Module

Let $\mathbb K$ be a field, $A= \mathbb K [x,y]$ and $ M = Ax + Ay$. prove that $M$ is NOT a free module!
1
vote
0answers
28 views

Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
2
votes
1answer
52 views

Example of a faithful irreducible module

Let $R$ be a non trivial simple ring. I am trying to show that there is a faithful irreducible left $R$-module. Is the ring $R$ considered as a left module over itself such a module? I think ...
1
vote
1answer
53 views

Dimension of dual module

Let $R$ be a finite dimensional semisimple $k$ algebra ($R$ is not necessarily commutative) and $M$ be an $R$-bimodule such that $M$ has finite dimension over $k$. Define ...
3
votes
1answer
83 views

“Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?
1
vote
1answer
29 views

Cotensor and counit?

If M is a C-bicomodule, then considering C as a $C$-bicomodule also, is $M \square_C C \cong C$, where $\square_C$ is the cotensor product in $^C\mathscr{M}^C$.
1
vote
1answer
26 views

Commutativity of comodules

If C is a cocommutative R-coalgebra, R is some commutative semi-simple artinian ring and A and B are C-bicomodules, then is $A\otimes_R B \cong B \otimes_R A$ as $R-modules$. However, this also true ...
0
votes
1answer
92 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 ...
1
vote
0answers
44 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 ...
2
votes
1answer
24 views

Smooth algebras and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
2
votes
1answer
95 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 ...
1
vote
0answers
62 views

Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
6
votes
0answers
152 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 & ...
3
votes
1answer
104 views

Problem with Wedderburn Theorem proof.

This is a Wedderburn Theorem proof in Frank W. Anderson, Kent R. Fuller: Rings and Categories of Modules Please explain that: "Therefore $_RR$ has a composition series of length $n$". Exercise ...
4
votes
1answer
57 views

Weak flat condition?

Let $R$ be a unit ring (not necessarily commutative). Then it is clear that for a right $R$-module $M$ we have: $M$ is flat $R$-module $\Rightarrow$ for any left $R$-module $E$ with $E\otimes_{R}M=0$ ...
1
vote
1answer
92 views

Is there a proof of this that does not use idempotents?

I am going to present a statement and a proof. The proof makes use of idempotents which makes it a little cumbersome. Is there a proof that does not use idempotents? (using well-known theorems ...
2
votes
2answers
137 views

Multiplicity of the simple $R$-module $M$ in the semisimple ring $R$

I'm confused about the conclusion of Wedderburn's structure theorem for semisimple rings. Let's consider the special case where $R=M^n$ as modules for some simple module $M$. Wedderburn's theorem says ...
3
votes
1answer
103 views

Question about Noncommutative Rings by I. N. Herstein

Here's an example that I do not quite understand it fully, it's on page 6 of the book. And here's what it says: Let $F$ be a field, and $F_n$ be a ring of all $n\times n$ matrices over $F$. We ...
2
votes
2answers
39 views

Partial cycles in projective resolutions of square-free algebra

Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand? I suspect not, but have not ...
2
votes
0answers
71 views

Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring ...
2
votes
1answer
112 views

An ideal generated by a nontrivial idempotent is not a free module

Let $R = \Bbb QG$ with $G$ isomorphic to a cyclic group of order $2$, and $x$ its generator. I'm trying to show that $$\frac{1}{2}(1+x)R$$ is not a free $R$-module. I found out that ...
2
votes
1answer
82 views

When do finiteness conditions coincide on modules?

Let $R$ be a (possibly noncommutative, but unital) ring. There are several finiteness conditions on a left $R$-module that we can consider. The ones I have come across include finitely generated ...
1
vote
1answer
102 views

Total divisor in a Principal Ideal Domain.

Let $R$ be a right and left principal ideal domain. An element $a\in R$ is said to be a right divisor of $b\in R$ if there exists $x \in R$ such that $xa=b$ . And similarly define left divisor. $a$ ...
0
votes
1answer
64 views

How to prove finite dimensionality of Hom-spaces between modules of finite length?

Let $R$ be a ring and $k$ be a field such that $k\hookrightarrow R$. Thus, given two $R$-modules $M$ and $N$, we can regard $\operatorname{Hom}(M,N)=\operatorname{Hom}_R(M,N)$ as a vector space over ...
3
votes
1answer
95 views

Natural homomorphism $\mathfrak{a} \prod E_\lambda \to \prod \mathfrak{a} E_\lambda$

Let $A$ be a ring (we don't assume that $A$ is commutative), and $\frak a$ a left ideal of $A$. Let $(E_\lambda)$ be a system of left $A$-modules. There is a natural homomorphism of modules $$ \phi ...
4
votes
0answers
240 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
3
votes
1answer
276 views

Trivial extension of an algebra

Suppose that $A$ is a finite dimensional $k$-algebra. Call $Q=\mathrm{Hom}_k(A,k)$. $Q$ admits an $A$-$A$-bimodule structire in the obvious way. The trivial extension of $A$ is defined as follows: ...
3
votes
1answer
117 views

Selfinjectivity and Frobenius algebras

Suppose that $R$ is a finite dimensional $k$-algebra. I say that $R$ is Frobenius if it is locally bounded (see this question for a definition) and indecomposable projectives and injectives coincide. ...
2
votes
1answer
69 views

On locally bounded algebras

Let $k$ be a field and $R$ an associative $k$-algebra suppose $R^2=R$. We say that $R$ is locally bounded if there exists a complete set of pairwise orthogonal primitive idempotents $\{e_x:x\in I\}$ ...