Studies in Rings generalised Unique Factorisation Rings

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Definition 1 054.
Let A be a" right R-module and B a submodule of A.
B is said to be an essential submodule of A if B ne 1= o.
for every non zero submodule C of A.
Definition 1.55 ..
Let Q be a ring. A right order in Q is any subring
R ~ Q such that
(a)
(b)
Every regular element of R is invertible in Q
-1
Every element of Q has the form ab for some
a ~ R and some regular element b in R.
It is clear that the ring Q in the definition 1055
and the localization of the ring R at the multiplicative
set CR(O)
are same.
Remark 1.56.
A right Goldie ring is any ring R,
such that R has
finite right rank and ACe on right annihilators. Thus
every righ.t Noe the r i an 4I'hg is righ t Goldie.
Remark 1057.
Goldie ha s proved that in a semiprime right Goldie ring
every essential. right ideal contains a
regular element and