Studies in Rings generalised Unique Factorisation Rings

-9-
Proposition 1~13.
For a proper prime ideal P in a ring R, the following
are equivalent.
(a") P is ~ prime ideal
(b)
Rip is a prime ring
(c) If xpy E R with xRy~P, either x £ P or y E: P
Cd) If I and J are any two right ideals of R such
that 1J ~ P, either I ~ P or J ~ P
(e) If I and J are any two left ideals such that
IJ ~ P, either I ~ P or J ~ P.
It follows immediately (by induction) from the above
Jl, ••. ,J n
are right (or left) ideals of R such that
3 1
J 2
•.. J < P, then some J. < P.
- n - 1 -
Proposition 1014.
Every maximal ideal M of a ring R is a prime ideal.
Definition 1.15.
A minimal prime ideal in a
ring R is any prime ideal
which does ~0t prope~ly contain any other prime ideal •
For instance, if R is a prime ring, then 0 is a
minimal prime ideal.