Prime ring
= Prime ring
{title2=$AB=0\Rightarrow A=0\text{ or }B=0$}
{wiki}
A prime ring is a nonzero <ring> whose zero ideal is a <prime ideal of a noncommutative ring>. Equivalently, the product of any two nonzero two-sided <ideals> is nonzero. This condition is weaker than absence of zero divisors: a full matrix ring over a <division ring> is prime even when its size is greater than one. For a commutative <ring>, primeness is exactly being an <integral domain>.