Studies in Rings generalised Unique Factorisation Rings

-52-
Proof:
As in [22, Proposition 2.10J.
Theorem 2.26.
prime GUFR s ,
Every semiprime GUFR is a sub direct product of
Proof:
Then
n
i=l P. 1
P be the minimal prime ideals of R.
n
= 0, since R
is semi prime. Thus, if we
show that RIp., for 1 ~ i ~ n, is a prime GUFR, then
1
the theorem follows from proposition 2.25.
It is clear
that Rip· is a prime Noetherian ring, for 1 $ i ~ n.
1
Let pip· be a non zero prime ideal of Rip.. Then P is a
1 1
non-minimal prime ideal of R with P. < Po So ther~
1 -
exists an element a e P such that aR=Ra is invertible.
By lemma 2.19, a ~ P. and thus a is a non zero element
1
o f RIP. with a(R/ P .) = (Rip· )a, ~Jhere a = a+P.. A1 so
1 l l 1
is the simple Artinian quotient ring of Rip·.
Now Rip.
1 1
is a prime GUFR follows from the fact that ~ £ pip ..
l
PRINCIPAL IDEAL RINGS
It is well known
that every commutative principal
ideal domain is a UFO. We prove an analogous result for
GUFRs.