0
votes
0answers
40 views

An error in the book “noncommutative ring” writed by Herstein

I'm reading the book "noncommutative ring" writed by Herstein. In the page 15, the author says that Let $F$ be a field and $A$ is an algebra over $F$. Let $\rho$ be a maximal regular right ideal ...
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
59 views

Locally unital ideals [duplicate]

Let $R$ be a ring with unity not necessarily commutative and $I$ an ideal of $R$. Let for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. ...
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 ...
0
votes
1answer
66 views

Direct product of algebras over a field

Let $ B_1,B_2,...,B_n$ k-algegras, $ B=\prod_{i=1}^{n}B_i $ the direct product of those (k is a field) , and $ J_i$ an ideal of its k-algebra. i must to prove that: The direct product $ ...
1
vote
1answer
118 views

Is every left maximal ideal the annihilator of a simple left module?

In my version of Noncommutative Algebra, by Benson Farb & R. Keith Dennis, in chapter I, section 2 on the Jacobson radical, it is claimed that … each maximal left ideal $I$ is the annihilator ...
2
votes
1answer
70 views

Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring. Now suppose that I have a non-commutative ring $N$. Suppose ...
0
votes
0answers
107 views

Rings, annihilators and (maximal) ideals

Let $R$ be a unital, associative, non-commutative ring. If $P$ is an ideal of $R$, what is the annihilator of quotients $R/PR$ and of $R/P$? Does something change if $P$ is supposed to be a maximal ...
1
vote
0answers
63 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 ...
1
vote
1answer
60 views

Clarification about some proof of Projectivity

Small provides an example of a ring which is right but not left hereditary is the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q} \end{matrix} \right)$; To ...
6
votes
1answer
80 views

P(R) is contained in Nil(R) for noncommutative rings.

How to show that $P(R)$ is contained in $\operatorname{Nil}(R)$ (where $R$ is a noncommutative ring with identity)? Definitions I am using: A nil right ideal is one whose elements are all ...
1
vote
1answer
110 views

No nonzero proper ideals of $K$-algebra $A$ implies the ring $A$ has no nonzero proper ideals

This is from Seth Warner's Classical Modern Algebra. The problem is: If $A$ is a nontrivial $K$-algebra possessing no nonzero proper ideals, then there are no nonzero proper ideals of the ring ...
1
vote
1answer
78 views

How to prove that every simple left $R$-module is isomorphic to a minimal left ideal of $R$

We know that: $T$ is a simple left $R$-module $\Longleftrightarrow T\cong R/M$, where $M$ is a maximal left ideal of $R$. So please tell me how to prove that every simple left $R$-module is ...
3
votes
1answer
152 views

Left and Right Ideal Generated by Two Matrices.

Let $R= {\rm Mat}_2(\Bbb R)$ be the ring (with $1$) of $2\times2$-matrices with entries in $\Bbb R$. Let $$M = \left\{\begin{pmatrix}1&0 ...
4
votes
2answers
80 views

$I$ semisimple + $R/I$ semisimple $\implies$ $R$ semisimple

Let $R$ be a (not necessarily commutative) ring with unit. Let $I\subset R$ be an ideal that in turn is a ring with unit. Is there a theorem that says something like $I$ semisimple and and $R/I$ ...
2
votes
1answer
46 views

In the lattice of ideals, what are the lowerbounds of the prime ideals?

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the ...
3
votes
2answers
251 views

Intersection of a subring and an ideal

Given a unital ring $R$ and its unital subring $P$ (with the same unit). Also, given a maximal left ideal $L$ of $R$. Must $L\cap P$ be a maximal left ideal of $P$? I don't think so, but are there any ...
1
vote
1answer
108 views

Commutants of commutative algebras

Let $W$ be a unital algebra and let $V$ be its maximal abelian subalgebra. Must the commutant $V^\prime$ of $V$ be commutative?
1
vote
1answer
303 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 ...
2
votes
1answer
79 views

Diamonds of ideals, part 3

I'd like to wrap up the line of questioning started first in this question and then continued in this question. The only variant left to try is: "How close can you get to the Diamond lattice ...
9
votes
1answer
240 views

A ring that has exactly 7 left ideals (T. Y. Lam)

Exercise 3.25 in Lam's First Course states: Let $R$ be a ring that has exactly seven nonzero left ideals. Prove that one of them is an ideal (i.e. left and right) and provide an example of such a ...
7
votes
1answer
151 views

Proper ideals generated by central ideals

Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly ...