Let $R$ be a noncommutative ring without identity.
Recall that a ring $R$ is said to be a prime ring if $aRb=(0), a,b\in R$ implies that $a=0$ or $b=0$.
I have to prove that
- $R$ is a prime ring
if and only if
- if $A,B$ are two-sided ideals of R and $AB=(0)$ then either $A=(0)$ or $B=(0)$.
The implication from 1. to 2. is easy to establish, but I have difficulty with the converse implication. I can't think of how this is true if $R$ is not unital. This is an exercise contained in the book "Noncommutative rings" by Herstein.