It is shown (Theorem 2) that a
semi-prime, left noetherian, left hereditary, two-sided Goldie ring is right
noetherian if and only if the right module (Q∕R) ⊕ R contains a copy of
every simple right R-module, where Q is the classical quotient ring of R.
Theorem 5 gives several necessary and sufficient conditions for a semi-prime
principal left ideal ring which is right Goldie to be a principal right ideal
ring. Among these is that R∕A must be artinian for every essential left ideal
A.
It is known that a two-sided noetherian semi-prime rlng is principal on
the left if and only if it is principal on the right. On the other hand, if one
drops the ascending chain condition on the right side of R, examples are
known of principal left ideal domains (p.1.i. domains) which are not right
principal. But, if we require that they be right Ore as well, things may be
better.
