Abstract
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) φ R contains a copy of every simple right iέ-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 ring 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.l.i. domains) which are not right principal. But, if we require that they be right Ore as well, things may be better.
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