Vol. 103, No. 2, 1982

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On hereditary rings and Noetherian V -rings

Karl Andrew Kosler

Vol. 103 (1982), No. 2, 467–473

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.

Mathematical Subject Classification
Primary: 16A52, 16A52
Secondary: 16A33
Received: 1 October 1980
Revised: 26 October 1981
Published: 1 December 1982
Karl Andrew Kosler