When every ideal is $\phi$-P-flat

Abstract

Let $R$ be a commutative ring with nonzero identity. An $R$-module $M$ is called $\phi$-P-flat if $x \in \Ann(s)M$ for every non-nilpotent element $s \in R$ and $x\in M$ such that $sx=0$. In this paper, we introduce and study the class of $\phi$-PF-rings, i.e., rings in which all ideals are $\phi$-P-flat. Among other results, the transfer of the $\phi$-PF-ring to the amalgamation is investigated. Several examples which delineate the concepts and results are provided.

Keywords

• $\phi$-flat
• $\phi$-P-flat
• $\phi$-PF-ring
• PF-ring
• PN-ring
• ZN-ring
• $\phi$-von Neumann regular ring
• trivial extension

References

1. [1] D. F. Anderson and A. Badawi, On $\phi$-Prüfer rings and $\phi$-Bézout rings, Houston J. Math 30 (2), 331–343, 2004.
2. [2] D. D. Anderson and M. Winders, Idealization of a module, J. Comm. Algebra 1 (1), 3–56, 2009.
3. [3] G. Artico and U. Marconi, On the compactness of minimal spectrum, Rend. Sem. Mat. Univ. Padova 56, 79–84, 1976.
4. [4] A. Badawi, On divided commutative rings, Comm. Algebra 27 (3), 1465–1474, 1999.
5. [5] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extensions defined by Prüfer condi- tions, J. Pure Appl. Algebra 214 (1), 53–60, 2010.
6. [6] G. W. Chang and H. Kim, Prüfer rings in a certain pullback, Comm. Algebra, to appear, 2022, doi: 10.1080/00927872.2022.2149766.
7. [7] F. Cheniour and N. Mahdou, When every principal ideal is flat, Port. Math. (N.S.), 70 (1), 51–58, 2011.
8. [8] F. Couchot, Flat modules over valuation rings, J. Pure Appl. Algebra 211 (1), 235– 247, 2007.

9. [9] M. D’Anna, A construction of Gorenstein rings, J. Algebra 306 (2), 507–519, 2006.
10. [10] M. D’Anna, C. A. Finocchiaro, and M. Fontana, Amalgamated algebras along an ideal, in: Commutative Algebra and Applications, Proceedings of the Fifth International Fez Conference on Commutative Algebra and Applications, Fez, Morocco, 2008, pp. 155-172, W. de Gruyter Publisher, Berlin, 2009.
11. [11] M. D’Anna, C. A. Finocchiaro, and M. Fontana, Properties of chains of prime ideals in amalgamated algebras along an ideal, J. Pure Appl. Algebra 214 (9), 1633–1641, 2010.
12. [12] M. D’Anna, C. A. Finocchiaro, and M. Fontana, New algebraic properties of an amalgamated algebra along an ideal, Comm. Algebra 44, 1836–1851, 2016.
13. [13] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443–459, 2007.
14. [14] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
15. [15] A. El Khalfi, H. Kim and N. Mahdou, Amalgamation extension in commutative ring theory, a survey, Moroccan Journal of Algebra and Geometry with Applications 1 (1), 139–182, 2022.
16. [16] S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics 1371, Berlin: Spring-Verlag, 1989.
17. [17] J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York Basel, 1988.
18. [18] S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (10), 3937–3953, 2004.
19. [19] H. Kim and F. Wang, On $\phi$-strong Mori rings, Houston J. Math. 38 (2), 359–371, 2012.
20. [20] J.J. Rotman, An Introduction to Homological Algebra, Springer, New York, 2009.
21. [21] F.Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications 22, Springer, Singapore, 2016.
22. [22] W. Zhao, On $\phi$-flat modules and $\phi$-Prüfer rings, J. Korean Math. Soc. 55 (5), 1221– 1233, 2018.
23. [23] W. Zhao, F. Wang and G. Tang, On $\phi$-von Neumann regular rings, J. Korean Math. Soc. 50 (1), 219–229, 2013.