Studies in Rings generalised Unique Factorisation Rings

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that the right ideal generated by a
right regular
element is right essential. As a consequence, in such
rings right regular elements are regular.
Also it can
be seen easily that any ideal in a
prime right Goldie
ring is essential as a right ideal and as a left ideal
and
so it contains regular elements.
Theorem 1.58 (Goldie)
A ring R is a right order in a semisimple Artinian
ring Q if and only if R is a semiprime right Goldie ring.
Theorem 1.59 (Goldie)
A ring R is a
right order in a simple Artinian ring
Q .if and only if R is a
prime right Goldie ring.
Remark 1060.
The ring Q, as' in theorem 1 058, is called a right
Goldie quotient ring of R.
Analogous results exist for
left semiprime (prime) Goldie ringso
When both left Goldie
quotient ring and right Goldie quotient ring exist they
can be identified and called ~e
Goldie quotient ring.
An important property of Q R
is that it will be the
injective hull of RR.