Studies in Rings generalised Unique Factorisation Rings

-51-
Let I be a non zero ideal of R. Let Pl,P 2
•.• Pm
be the minima 1 primes over I (relna rk 1.18). Then the
produe t P1P2 0.. p m ~ I. By hYpoth e s i S t for 1 $ i $ m ,
th ere ex i s t s -a. E:
1
R s uch that a,R = Ra. < P. and each
1 1 - 1
a.R = Ra., for 1 5 i ~ rn, is invertible. Then
1 1
a la2
0 •• amR = Ral···a rn 5 P 1
invertible.
Now let I and J be two non zero ideals of Ro It
follows by the above paragraph that there exists a €
I
and b ~ J such that aR=Ra and bR=Rb and they are invertible.
Thus, by lemma 2 06, both a and bare r-e qul a r . Consequently
o 1= ab € IJ and we have IJ 1= o. Th e r e f o r e the product of
two nqnzero ideals of R is non zero, which implies R is prime.
A ring R is said to be a
sub direct product of the
rings { Si/i E: J} if there is a monomorph i sm
K: R -~ S = 1t s. (the direct product of S.s) such
itJ 1 1
that n.oK is surjective for all i,
1
the natural projection.
wh e re 'Tt • :
1
S ----1 S. is
1
Proposition 2 025.
if S.
R is a sub direct product of Si' i E IJif and. only
is isomorphic to R/K., where K.s are ideals of R
l 1 1
with n K. = O.
i E I J.