# Almost valuation rings

Document Type: Research Paper

Authors

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.

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Main Subjects

### References

D. F. Anderson, Comparability of ideals and valuation overrings, Houston J. Math. 5 (1979), no. 4, 451--463.

D. F. Anderson, When the dual of an ideal is a ring, Houston J. Math. 9 (1983), no. 3, 325--332.

D. F. Anderson, A. Badawi and D. E. Dobbs, Pseudo-valuation rings, II, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 3 (2000), no. 2, 535--545.

D. F. Anderson and D. E. Dobbs, Pairs of rings with the same prime ideals, Canad. J. Math. 32 (1980), no. 2, 362--384.

D. D. Anderson and M. Zafrullah, Almost Bezout domains, J. Algebra 142 (1991), no. 2, 285--309.

A. Badawi, On domains which have prime ideals that are linearly ordered, Comm. Algebra 23 (1995), no. 12, 4365--4373.

A. Badawi, Remarks on pseudo-valuation rings, Comm. Algebra 28 (2000), no. 5, 2343--2358.

A. Badawi, Pseudo-valuation domains: A survey, Proceedings of the Third Palestinian International Conference on Mathematics, (Bethlehem, 2000), pp. 38-59, World Scientific, River Edge, NJ, 2002.

A. Badawi, On pseudo-almost valuation domains, Comm. Algebra 35 (2007), no. 4, 1167--1181.

A. Badawi, D. F. Anderson and D. E. Dobbs, Pseudo-valuation rings, in: Commutative Ring Theory (F_es, 1995), pp. 57--67, Lecture Notes in Pure Appl. Math. 185, Marcel Dekker, New York, 1997.

A. Badawi and E. G. Houston, Powerful ideals, strongly primary ideals, almost pseudovaluation domains, and conducive domains, Comm. Algebra 30 (2002), no. 4, 1591--1606.

J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Paci_c J. Math. 75 (1978), no. 1, 137--147.

J. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York-Basel, 1988.

R. Jahani-Nezhad and F. Khoshayand, Pseudo-almost valuation rings, Bull. Iranian Math. Soc. 41 (2015), no. 4, 815--824

N. Mahdou, A. Mimounui and M. A. S. Moutui, On almost valuation and almost Bezout rings, Comm. Algebra 43 (2015), no. 1, 297--308.

### History

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