Prime ring

In abstract algebra, a non-trivial 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. Prime ring can also refer to the subring of a field 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).^[1]

Prime rings, under the first definition, can be regarded as a simultaneous generalization of both integral domains and matrix rings over fields.

Examples

• Any domain is a prime ring.
• Any simple ring is a prime ring, and more generally: every left or right primitive ring is a prime ring.
• Any matrix ring over an integral domain is a prime ring. 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.
• A ring is prime if and only if its zero ideal is a prime ideal.
• A non-trivial ring is prime if and only if the monoid of its ideals lacks zero divisors.
• The ring of matrices over a prime ring is again a prime ring.

Notes

1. ^ Page 90 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0 

References

• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0, MR1838439