Studies in Rings generalised Unique Factorisation Rings

-74-
Now, if x ( P, then xS = Sx is contained in P
and xS = Sx is Q(S)-invertible. Otherwise, we prove
that as = Sa
is Q(S)-invertible, where a is as in the
above pareqra ph , Let 9 E: S and assume 9=cofc1 x+~. _+cmx m
whe re c. £ R, for 0 ~ i ~ n . Sinc e a (aR) = aR, i t
1
follows that a(a) = au, for some unit u in Ro Consider
+ C xffi)a = c a + c1xa + ••• +c xffia
ID 0 m
• •• + c a n1( a ) xm
m
[where d. =
1
i-I
n aj(~), where n stands for product]
j=O
Thus Sa ~ as and similarly as ~ Sa. Also a € P and the
proof is complete as as = Sa
is Q(S)-invertible (since
Next assume that S is prime. Then S has a simple
Artinian quotient ring Q(S) by Goldie's theorem. In this
case,
the proof is similar to the proof given in
[2, theorem 4.1J, we sketch ito