Studies in Rings generalised Unique Factorisation Rings

Necessity
Since S is a prime ring it follows that R is
a-prime. Let P be a non-zero a-prime ideal of Ro
Because PS is a non-zero prime ideal of S, there is a
non-zero element 9 of PS such that 95 = 59 is 0(5)­
invertible, where Q(5)
is the simple Artinian quotient
ring of S. Clearly x ~ PS o For, if x £ PS, then
n
x = L r , f. , where r.E P and f. e S, ( 1)
1 1 1 1
i=o
for 0 S i ~ n.
Equating the coefficient on both sides of (1) we get
and the remaining coefficients in R.H.S of (1) vanish.
k.
(Here we are assuming that f. = a, + a1'lx + •.. + a' k
x 1
1 10 1 . 1
for each i and k i
is a non-negative integer). But each
r i
E: P, for 0 ~ i ~ n , implies that 1 £ P. Thus g 1= x,
and
without loss of generality we may assume that
n
9 = Co + clx + 0 •• + cnx , where c i
£ P for ea eh i.
Since g5 = 59, we have gR = Rg and since 9 is regular
in S, 9 is regular in R. Now c.R =
1
Rc. for each i.
1
For, let r e R, then a-i(r) f: Ro Since gR = Rg, there