A two-sided ideal of a ring such that the inclusion for any two-sided ideals and of implies that either or . An ideal of a ring is prime if and only if the set is an -system, i.e. for any there exists an such that . An ideal of a ring is prime if and only if the quotient ring by it is a prime ring.
This assumes that the empty set is an m system by default.
|[a1]||L.H. Rowen, "Ring theory" , I , Acad. Press (1988) pp. 163|
Prime ideal. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Prime_ideal&oldid=29171