Bulletin of the Iranian Mathematical Society

Bulletin of the Iranian Mathematical Society

Almost valuation rings

Document Type : Research Paper

Authors
R. Jahani-Nezhad 1
F. Khoshayand 2
1 Department of‎ ‎Mathematics‎, ‎University‎ ‎of Kashan‎, ‎P.O‎. ‎Box 8731751167‎, ‎Kashan‎, ‎Iran.
2 Department of‎ ‎Mathematics‎, ‎University‎ ‎of Qom‎, ‎P.O‎. ‎Box 3716146611‎, ‎Qom‎, ‎Iran.
Abstract
The aim of this paper is to generalize the‎ ‎notion of almost valuation domains to arbitrary commutative‎ ‎rings‎. ‎Also‎, ‎we consider relations between almost valuation rings ‎and pseudo-almost valuation rings‎. ‎We prove that the class of‎ ‎almost valuation rings is properly contained in the class of‎ ‎pseudo-almost valuation rings‎. ‎Among the properties of almost‎ ‎valuation rings‎, ‎we show that a quasilocal ring $R$ with regular‎ ‎maximal ideal $M$ is a pseudo-almost valuation ring if and only‎ ‎if $V = (M‎ : ‎M)$ is an almost valuation ring with maximal ideal‎ ‎${\rm Rad}_V(M)$‎. ‎Furthermore‎, ‎we show that pseudo-almost valuation‎‎rings are precisely the pullbacks of almost valuation rings‎.
Keywords
Strongly prime ideal
Almost valuation domain
Pseudo-almost valuation ring

Subjects

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Bulletin of the Iranian Mathematical Society
Volume 43, Issue 3
June 2017
Pages 807-816

  • Receive Date 14 April 2015
  • Revise Date 26 February 2016
  • Accept Date 26 February 2016