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 the prime ring is the finite field of order p.Equivalent definitions
A ring R is prime if and only if the zero ideal 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:
- For any two ideals A and B of R, AB = implies A = or B =.
- For any two right ideals A and B of R, AB = implies A = or B =.
- For any two left ideals A and B of R, AB = implies A = or B =.
Using these conditions it can be checked that the following are equivalent to R being a prime ring:
