Studies in Rings generalised Unique Factorisation Rings

-12-
(a) I is a semiprime ideal
(b) If J is any ideal such that J2 S I, then J ~ I.
Corollary 1 022 0
Let I be a semiprime ideal in a ring R, J be any
left or right ideal of R such that In ~ I for some
positive integer n, then J 5 I.
Definition 1.23.
A right or left ideal J in a ring R is nilpotent
provided In = 0 for some positive integer no More generally,
J is nil provided every element of J is nilpotent.
Definition 1.24.
The prime radical of a
ring R is the intersection of
all prime ideals of R.
It is easy to observe that the prime radical of any
ring is nil and R is semiprime if and only if its prime
radical is zero.
proposition 1.25.
In any ring R,
the prime radical equals the intersection
of all minimal prime ideals.