Prime ring

In abstract algebra, a nonzero ring "R" is a prime ring if for any two elements "a" and "b" of "R", "arb = 0" for all "r" in "R" implies that either "a = 0" or "b = 0". This definition can be regarded as a simultaneous generalization of both integral domains and simple rings.
Although this article discusses the above definition, prime ring may also refer to the minimal non-zero subring of a field, which is generated by its identity element 1, and determined by its characteristic. For a characteristic 0 field, the prime ring is the integers, for a characteristic "p" field (with "p" a prime number) the prime ring is the finite field of order "p" (cf. prime field).
Equivalent definitions.
A ring "R" is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense.
This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for "R" to be a prime ring:
Using these conditions it can be checked that the following are equivalent to "R" being a prime ring: