Studies in Rings generalised Unique Factorisation Rings

-60-
RI~JGS yVITli [::NOUGH INVER~rIBI.JE ILJEALS
INerecall t h ()tari n 9 R i s s aid t 0 be righ t (1eft)
hereditary, if all of its right (left) ideals are
projective. If f{ is No e t.h e r i an , then 1\ is left hereditary
if and only if R is right hereditary [6, corollary 8018J
and
in this case R is called hereditary.
Definition 2.36.
If every essential right ideal of a ring R contains
a
non zero ideal, then R is said to be right bounded. By
symmetry we define left bounded rings and R is said to
be bounded if it is both left and r i qh t bounded.
Definition 2.37.
If every non zero ideal of a ring R contains an
invertible ideal, then R is said to be a ring with enough
invertible ideals.
We
state some results that are given in CS].
Lemrna
2. 38 •
If R is a
right bounded hereditary Noetherian prime
Ring,
then R has enough invertible ideals.
Lemma 2039.
If R h~s
enough invertible ideals, then R is bounded
or p r cmi t i v e ,