In abstract algebra, a ring R is a prime ring if for any two elements a and b of R, if arb = 0 for all r in R, then either a = 0 or b = 0.

Prime rings can be regarded as a simultaneous generalization of both integral domains and matrix rings over fields.

Examples

• Any domain.
• Any primitive ring.
• A matrix ring over an integral domain. In particular, the ring of 2-by-2 integer matrices is a prime ring.

Properties

• A commutative ring is a prime ring if and only if it is an integral domain.
• The ring of matrices over a prime ring is a prime ring.
• A ring is prime if and only if its zero ideal is a prime ideal.