Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON Φ-FLAT MODULES AND Φ-PRÜFER RINGS

  • Zhao, Wei (School of Mathematics and Computer Science ABa Teachers university)
  • Received : 2017.10.18
  • Accepted : 2018.02.06
  • Published : 2018.09.01
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Abstract

Let R be a commutative ring with non-zero identity and let NN(R) = {I | I is a nonnil ideal of R}. Let M be an R-module and let ${\phi}-tor(M)=\{x{\in}M{\mid}Ix=0\text{ for some }I{\in}NN(R)\}$. If ${\phi}or(M)=M$, then M is called a ${\phi}$-torsion module. An R-module M is said to be ${\phi}$-flat, if $0{\rightarrow}{A{\otimes}_R}\;{M{\rightarrow}B{\otimes}_R}\;{M{\rightarrow}C{\otimes}_R}\;M{\rightarrow}0$ is an exact R-sequence, for any exact sequence of R-modules $0{\rightarrow}A{\rightarrow}B{\rightarrow}C{\rightarrow}0$, where C is ${\phi}$-torsion. In this paper, the concepts of NRD-submodules and NP-submodules are introduced, and the ${\phi}$-flat modules over a ${\phi}-Pr{\ddot{u}}fer$ ring are investigated.

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References

  1. D. F. Anderson and A. Badawi, On ${\phi}$-Prufer rings and ${\phi}$-Bezout rings, Houston J. Math. 30 (2004), no. 2, 331-343.
  2. D. F. Anderson and A. Badawi, On ${\phi}$-Dedekind rings and ${\phi}$-Krull rings, Houston J. Math. 31 (2005), no. 4, 1007-1022.
  3. K. Bacem and B. Ali, Nonnil-coherent rings, Beitr. Algebra Geom. 57 (2016), no. 2, 297-305. https://doi.org/10.1007/s13366-015-0260-8
  4. A. Badawi, On divided commutative rings, Comm. Algebra 27 (1999), no. 3, 1465-1474. https://doi.org/10.1080/00927879908826507
  5. A. Badawi, On ${\phi}$-pseudo-valuation rings, in Advances in commutative ring theory (Fez, 1997), 101-110, Lecture Notes in Pure and Appl. Math., 205, Dekker, New York, 1999.
  6. A. Badawi, On ${\Phi}$-pseudo-valuation rings. II, Houston J. Math. 26 (2000), no. 3, 473-480.
  7. A. Badawi, On ${\phi}$-chained rings and ${\phi}$-pseudo-valuation rings, Houston J. Math. 27 (2001), no. 4, 725-736.
  8. A. Badawi, On divided rings and ${\phi}$-pseudo-valuation rings, in Commutative rings, 5-14, Nova Sci. Publ., Hauppauge, NY, 2002.
  9. A. Badawi, On nonnil-Noetherian rings, Comm. Algebra 31 (2003), no. 4, 1669-1677. https://doi.org/10.1081/AGB-120018502
  10. A. Badawi, Factoring nonnil ideals into prime and invertible ideals, Bull. London Math. Soc. 37 (2005), no. 5, 665-672. https://doi.org/10.1112/S0024609305004509
  11. A. Badawi, On rings with divided nil ideal: a survey, in Commutative algebra and its applications, 21-40, Walter de Gruyter, Berlin, 2009.
  12. A. Badawi and D. E. Dobbs, Strong ring extensions and ${\phi}$-pseudo-valuation rings, Houston J. Math. 32 (2006), no. 2, 379-398.
  13. A. Badawi and T. G. Lucas, Rings with prime nilradical, in Arithmetical properties of commutative rings and monoids, 198-212, Lect. Notes Pure Appl. Math., 241, Chapman & Hall/CRC, Boca Raton, FL, 2005.
  14. A. Badawi and T. G. Lucas, On ${\Phi}$-Mori rings, Houston J. Math. 32 (2006), no. 1, 1-32.
  15. D. E. Dobbs, Divided rings and going-down, Pacific J. Math. 67 (1976), no. 2, 353-363. https://doi.org/10.2140/pjm.1976.67.353
  16. L. Fuchs and L. Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, 84, American Mathematical Society, Providence, RI, 2001.
  17. M. Griffin, Prufer rings with zero divisors, J. Reine Angew. Math. 239/240 (1969), 55-67.
  18. S. Hizem and A. Benhissi, Nonnil-Noetherian rings and the SFT property, Rocky Mountain J. Math. 41 (2011), no. 5, 1483-1500. https://doi.org/10.1216/RMJ-2011-41-5-1483
  19. J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
  20. H. Kim and F. Wang, On ${\phi}$-strong Mori rings, Houston J. Math. 38 (2012), no. 2, 359-371.
  21. C. Lomp and A. Sant'Ana, Comparability, distributivity and non-commutative ${\phi}$-rings, in Groups, rings and group rings, 205-217, Contemp. Math., 499, Amer. Math. Soc., Providence, RI, 2009.
  22. F. G. Wang, Commutative Rings and Star-Operation Theory, Sicence Press, Beijing, 2006.
  23. R. B. Warfield, Jr., Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25 (1970), 167-172. https://doi.org/10.1090/S0002-9939-1970-0254030-4
  24. W. Zhao, F. Wang, and G. Tang, On ${\varphi}$-von Neumann regular rings, J. Korean Math. Soc. 50 (2013), no. 1, 219-229. https://doi.org/10.4134/JKMS.2013.50.1.219