Studies in Rings generalised Unique Factorisation Rings

-84-
each j which is not possible. An argument using "prime
avoidanc e" theorem again shows tha t RPl RP2R ~ U.
Thus there exists at least one element p € RPl RP2R such
tha t p rt U. Sinc e pERP1 RP2R, P Go P1 () P2 • Ne x t
consider P3 and proceed as above, we get an element
pi £: Rp RP3R such that pi Eo U, also pi c Rp RP3R such
that p ' € U, also p ' e: PIr) P 2('\
P 3•
Continue t.he process
until all the P.'s exhausted, we get an element
1
n
u E n P. such tha t u f.
.1 1 1=
m
(J Q .•
j=l J
Defin i t ion 3 0 27 •
Let R b~ a ring and S an over-ring of R. Then a
weakly S-invertible element in R is any element a
in R
n
such that 1 = E a.ab. for some a.,b. in S, for l$i$no
i=l 1 1 1 1
Equivalently the ideal SaS = s.
Example s 3.28.
(1) Every unit in a ring R is weakly R-invertible.
(2) If R is a prime Noetherian ring with the
simple Artinian quotient ring OCR), Q(R)aQ(R)=Q(R)
for any 0 f:. a ~ R; Thus every non zero element
in R
is weakly Q(R)-invertible.