Studies in Rings generalised Unique Factorisation Rings

-78-
is an r' E
R such that
Equa t ilng th e 1· th coe f f ilClen c i t s,we ge t
c.r = r'c., which implies c.R < Rc.
1 1 . 1 - ].
and similarly
Rc. < c.R. This observation together with the fact that
1 - 1
9 is regular in R implies that c. is a regular element
1
of R, for some i, 0 5 i ~ n.
Next we prove c.R = Rc. is an a-ideal of R. We
1 1
consider a(r)x (co+c1x + •.. + cnx n) = (co+c1x+ •••+cnxn)r'x
f or some r I ~ R 0 Eoua qua t ilng th e 1 · th . t erm coe f f 1clen i · t S 0 f
this expres~~on we get a(r) a(c.) = c.a~r'), i.eo
1 J.
a(rc.) = c.a:tr') and hence a(c.R) = a:(Rc.) < c.R.
1 ]. 1]. .. 1
Thus the non zero~prime ideal P contains at least
one regular element c. such that c.R = Rc. is an a-ideal
]. J. J.
and c.R = Rc . is Q(R)-invertible, since c. € CR(O). This
1 1. l.
completes the proof.