Studies in Rings generalised Unique Factorisation Rings

-8-
In non-commutative rings, it turns out that it is
not a
good idea to concentrate on prime ideals P such that
RIp is a domain (ab E P implies a f: P or b E p). In f a c t ,
there are many non-commutative rings with no factor rings
which are domains. Thus the desirable thing is to give
a more relaxed definition for prime idealso The key is
to change the commutative definition by replacing products
of elements by products of ideals which was
first proposed
by Krull in 1928.
Definition 1.12.
A prime ideal in a ring R is a proper ideal P of R
such that whenever I and J are ideals of R with IJ ~ P,
either I ~ P or J ~ P, P is said to be a completely prime
ideal, if whenever a,b t R such that ab E P, either a c: P
or b £ P. A prime ring is a ring in which 0 is a prime
ideal and a domain is a ring in which 0
is a completely
prime ideal.
From part (c)
of the following proposition it follows
t hat in th e co mmuta t i v e case th e pri me i de a 15 and th e
completely prime ideals coincide with the usual prime ideals
and
is a
in non-commutative setting, every completely prime ideal
prime ideal.