Studies in Rings generalised Unique Factorisation Rings

-113-
Artinian ring, more precisel~R is a subdirect product
of Lrr educ i b I.e
No e th e r i an rings, i. e. rings with Artinian
quotient ringso
In theorem 2 035, to prove that R[x] is a GUFR, we
assumed that E p < CR[x] (0) () Cl for every minimal prime
ideal P of R. We do not know whether this condition can
be relaxed. However, other than for prime rings R, no
examples of R[x]sJwith non mi n i rnn I
prime ideal. P.lcould
be found out with the property that, ~
P () R is a
mi n i malpr i me i d ea 1 i n R.
We
shall state a result given in [29, pp. 59-60J.
Lemma 5.1.
Let R be a right order in Q and A R
a submodule of
Q R that contains a regular element of R. Then A R is a
projective if and only if there exist elements Yl ..• Y n
in Q and al ..• an lh A such that Yi A 5 R for all i and
1 = a1Yl + a 2 Y2 + ••• + any n .
Now if R is a right bounded prime GUFR and I is
an essential right ideal of ~, then I contains an ideal J,
which in turn contains a normal ideal aR = Ra (say) of R