Studies in Rings generalised Unique Factorisation Rings

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Definition 1 0 4 9 .
A regul~r
element in R is any non-zero divisor,
i.e., any x~ R such that r(x) = 0 and L(x) = o.
Note that if R ~ Q are rings and x is any element
of R which is invertible in Q, then x is a regular element
in R.
Definition 1 050.
Let I be an ideal of R. An element x ~R is said
to be regular modulo I provided the coset x+I is regular
in R!l. The set of such x is denoted by e(l). Thus the
set of regular elements in R may be
we use the notation CR(I) for C(I).
denoted by CR(O). Often
Definition 1 051.
A right (left) annihilator ideal in a ring R is any
right (left) ideal of R which equals the right (left)
annihilator of some subset x.
Definition le52.
A ring R is said to be
of finite right (left) rank
if RR(RR) contains no infinite direct sum of submodules.
Definition 1 053.
A
ring R is said to be right (left) Goldie if RR(RR)
has finite rank and R has ACC on right (left) annihilators.