Research ArticleNo Access

Maximal non- $ϕ$ -pseudo-valuation rings

Rahul Kumar

Department of Mathematics, University of Delhi, Delhi, India

E-mail Address: rahulkmr977@gmail.com

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and

Atul Gaur

Department of Mathematics, University of Delhi, Delhi, India

E-mail Address: gaursatul@gmail.com

Corresponding author.

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https://doi.org/10.1142/S0219498822500906Cited by:1 (Source: Crossref)

Abstract

The notion of maximal non- $ϕ$ -pseudo-valuation ring is introduced which generalizes the concept of maximal non-pseudo-valuation domain. The equivalence of maximal non- $ϕ$ -PVR and maximal non-local ring is established under some conditions. The concept of maximal non-PVR is also introduced and it is shown that a maximal non-PVR is either a $ϕ$ -PVR or a maximal non- $ϕ$ -PVR. Examples are also given to show that all the introduced concepts are independent of each other, in general.

Communicated by B. Olberding

Keywords:

AMSC: 13B99, 13B22

References

1. D. F. Anderson and A. Badawi , On $ϕ$ -Prüfer rings and $ϕ$ -Bezout rings, Houston J. Math. 30(2) (2004) 331–343. Web of Science, Google Scholar
2. D. F. Anderson and A. Badawi , On $ϕ$ -Dedekind rings and $ϕ$ -Krull rings, Houston J. Math. 31(4) (2005) 1007–1022. Web of Science, Google Scholar
3. D. D. Anderson and M. Winders , Idealization of a module, J. Commut. Algebra 1(1) (2009) 3–56. Crossref, Web of Science, Google Scholar
4. A. Badawi , On divided commutative rings, Comm. Algebra 27(3) (1999) 1465–1474. Crossref, Web of Science, Google Scholar
5. A. Badawi , On ϕ-Pseudo-Valuation Rings, Lecture Notes in Pure and Applied Mathematics, Vol. 205 (Marcel Dekker, New York/Basel, 1999), pp. 101–110. Google Scholar
6. A. Badawi , On $ϕ$ -pseudo-valuation rings II, Houston J. Math. 26(3) (2000) 473–480. Web of Science, Google Scholar
7. A. Badawi , On $ϕ$ -chained rings and $ϕ$ -pseudo-valuation rings, Houston J. Math. 27(4) (2001) 725–736. Web of Science, Google Scholar
8. A. Badawi , On divided rings and $ϕ$ -pseudo-valuation rings, Int. J. Comm. Rings (IJCR) 1(2) (2002) 51–60. Google Scholar
9. A. Badawi , On nonnil-Noetherian rings, Comm. Algebra 31(4) (2003) 1669–1677. Crossref, Web of Science, Google Scholar
10. A. Badawi , Factoring nonnil ideals into prime and invertible ideal, Bull. London Math. Soc. 37(5) (2005) 665–672. Crossref, Web of Science, Google Scholar
11. A. Badawi, D. F. Anderson and D. E. Dobbs , Pseudo-Valuation Rings, Commutative Ring Theory, Lecture Notes in Pure and Applied Mathematics, Vol. 185 (Marcel Dekker, New York/Basel, 1997), pp. 57–67. Google Scholar
12. A. Badawi and A. Jaballah , Some finiteness conditions on the set of overrings of a $ϕ$ -ring, Houston J. Math. 34(2) (2008) 397–408. Web of Science, Google Scholar
13. A. Badawi and T. G. Lucas , Rings with prime nilradical, in Arithmetical Properties of Commutative Rings and Monoids, Vol. 241 (Chapman & Hall/CRC, 2005), pp. 198–212. Google Scholar
14. A. Badawi and T. G. Lucas , On $ϕ$ -Mori rings, Houston J. Math. 32(1) (2006) 1–32. Web of Science, Google Scholar
15. P. J. Cahen , Couples danneaux partageant un id $é$ al, Arch. Math. 51(6) (1988) 505–514. Crossref, Web of Science, Google Scholar
16. D. E. Davis , Overrings of commutative rings III: Normal pairs, Trans. Amer. Math. Soc. 182 (1973) 175–185. Web of Science, Google Scholar
17. P. A. Froeschl III , Chained rings, Pacific J. Math. 65(1) (1976) 47–53. Crossref, Web of Science, Google Scholar
18. A. Gaur and R. Kumar , Maximal non $ϕ$ -chained rings and maximal non chained rings, Results Math. 74(3) (2019), Article ID 121. Crossref, Web of Science, Google Scholar
19. R. Gilmer and W. Heinzer , Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7 (1967) 133–150. Crossref, Google Scholar
20. M. Griffin , Prüfer rings with zero divisors, J. Reine Angew. Math. 239/240 (1969) 55–67. Web of Science, Google Scholar
21. J. R. Hedstrom and E. G. Houston , Pseudo-valuation domains, Pacific J. Math. 75(1) (1978) 137–147. Crossref, Web of Science, Google Scholar
22. J. Huckaba , Commutative Rings with Zero Divisors (Marcel Dekker, New York/Basel, 1988). Google Scholar
23. N. Jarboui and S. Trabelsi , Some results about proper overrings of pseudo-valuation domains, J. Algebra Appl. 15(5) (2016) 1650099. Link, Web of Science, Google Scholar
24. M. Nagata , Local Rings (Interscience, New York, 1962). Google Scholar
25. M. B. Nasr and N. Jarboui , On maximal non-valuation subrings, Houston J. Math. 37(1) (2011) 47–59. Web of Science, Google Scholar