Answer to Question #101847 in Abstract Algebra for Maha Alghamdi

Question

Prove or disprove:
a) every left simple ring is right simple
b) every simple ring is left simple

Answer 1

a) A ring $R$ with 1 is called simple if $(0)$ and $R$ are the only two-sided ideals of $R$ .

If $R$ is a Artinian ring then left simple ring is right simple. By the Artin-Wedderburn theorem, a ring $R$ is simple and (left or right) Artinian if and only if $R=M_{n}(R)$ ( $M_{n}(R)$ be the ring of $n\times n$ matrices with entries from $R$ ) for some division ring $R$ and some integer $n\geq1$ .

$R$ every ring that the statement is incorrect because

subject to the problem $R$ every left simple ring

$\forall i\in I, \forall r\in R, ri\in I$ where $I$ ideals of $R$ , $I$ is not $(0)$ and $R$

$ir\not \in I$ in the general case.

b) Every simple ring is left simple is correct. A ring $R$ with 1 is called simple if (0) and $R$ are the only two-sided ideals of $R$ .

Subject to the problem $R$ every simple ring

$\forall i\in I, \forall r\in R, ri\in I, ir\in I$ where $I$ ideals of $R$ , $I$ is not $(0)$ and $R$ . Whereas $ri\in I$ $R$

is left simple ring