The purpose of this paper is
to examine conditions under which (1) a left noetherian left V -ring is left
hereditary and (2) a left noetherian left V -ring is a two sided noetherian
V -ring. For (1), left noetherian left V -rings which satisfy the restricted left
minimum (RLM) condition are examined. The RLM condition is shown to be
equivalent to E(R)∕R a semisimple left R-module. Consequently, hereditary is
equivalent to E(R)∕R semisimple in the two sided case. Two sided noetherian
V -rings which are critically nice are also examined. In this case, hereditary is
shown to be equivalent to E(R)∕R injective and smooth. For (2), a theorem
of Faith’s concerning left QI-domains is extended to left noetherian left
V -rings.
