-2
votes
0answers
23 views

locally unital ideals of ringa [duplicate]

1 down vote favorite Let R be a ring with unity not necessarily commutative and I an ideal of R.Let for every element a of I there exists an element c of I such that ac=a.Note that c is related to a. ...
3
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
42 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
64 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
86 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
61 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
100 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
59 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
59 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
109 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
76 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
141 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
244 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
107 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
287 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
78 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
234 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 ...