prime ring

Noun

 * 1)   Any nonzero ring R such that for any two (two-sided) ideals P and Q in R, the product PQ = 0 (the zero ideal) if and only if P = 0 or Q = 0.
 * 2)   A ring which is equal to its own prime subring.
 * 1)   A ring which is equal to its own prime subring.
 * 1)   A ring which is equal to its own prime subring.
 * 1)   A ring which is equal to its own prime subring.
 * 1)   A ring which is equal to its own prime subring.
 * 1)   A ring which is equal to its own prime subring.

Usage notes
The following conditions are equivalent to R being a prime ring :
 * for arbitrary a, b ∈ R, if arb = 0 for all r ∈ R (i.e., if aRb = 0) then either a = 0 or b = 0;
 * the zero ideal is a prime ideal in R.

and are not equivalent; for example, $$\mathbb{Z}[x]$$ is a prime ring in the sense of, but not in the sense of.