Studies in Rings generalised Unique Factorisation Rings

-54-
Theorem 2.28.
Let R be a Noetherian ring in which the principal
left ideals generated by regular elements are also
principal right idealso Then R is a GUFR if and only
if R has an Artinian quotient ring.
Proof:
Assume that R has an Artinian quotient ring Q(R).
Let P be a non-minimal prime ideal of Ro Then,pnCR(O)~,
by propos i tion 1.63. Let b € P n C (0), then by hypo thesis
R
there exists an element a € R such that Rb = aRe Since
R has an Artinian quotient ring, every left regular element
is "regular (Proposition 1.62). Hence, by lemma ~o27, we
ha ve a E: C R
(0 ) and a R = Ra . Sinc e a -1 (' Q ( R), a R = Ra
is Q(R)-invertibleo Also P contains aR = Ra. This
completes the proof of the sufficient part.
The necessary.
part of the theorem follows from
theorem 2. ·~'9.
Corollary 2.29.
Suppose R is a Noetherian ring in which every regular
element is normal. Then R is a GUFR if and only if R has
an Artinian quotient ring.