Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON SOME GENERALIZATIONS OF THE REVERSIBILITY IN NONUNITAL RINGS

  • Received : 2018.01.06
  • Accepted : 2018.07.05
  • Published : 2019.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.

Keywords

References

  1. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852. https://doi.org/10.1080/00927879908826596
  2. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Aust. Math. Soc. 18 (1974), 470-473. https://doi.org/10.1017/S1446788700029190
  3. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Aust. Math. Soc. 2 (1970), 363-368. https://doi.org/10.1017/S0004972700042052
  4. G. F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebra 11 (1983), no. 6, 567-580. https://doi.org/10.1080/00927878308822865
  5. W. Chen, On semiabelian $\pi$-regular rings, Int. J. Math. Math. Sci. 2007 (2007), Art. ID 63171, 10 pp. https://doi.org/10.1155/2007/63171
  6. P. M. Cohn, Reversible rings, Bull. Lond. Math. Soc. 31 (1999), no. 6, 641-648. https://doi.org/10.1112/S0024609399006116
  7. J. M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ. 32 (1990), 73-76.
  8. J. Han, Y. Lee, and S. Park, Semicentral idempotents in a ring, J. Korean Math. Soc. 51 (2014), no. 3, 463-472. https://doi.org/10.4134/JKMS.2014.51.3.463
  9. V. K. Harchenko, T. J. La ey, and J. Zemanek, A characterization of central idempotents, Bull. Acad. Pol. Sci. Ser. Sci. Math. 29 (1981), no. 1-2, 43-46.
  10. I. Kaplansky, Rings of operators. Notes prepared by S. Berberian with an appendix by R. Blattner, Mathematics 337A, summer 1955.
  11. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. https://doi.org/10.1006/jabr.1999.8017
  12. N. K. Kim, Y. Lee, and Y. Seo, Structure of idempotents in rings without identity, J. Korean Math. Soc. 51 (2014), no. 4, 751-771. https://doi.org/10.4134/JKMS.2014.51.4.751
  13. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. https://doi.org/10.4153/CMB-1971-065-1
  14. G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2003), no. 2, 494-520. https://doi.org/10.1016/S0021-8693(03)00301-6
  15. L. Motais de Narbonne, Anneaux semi-commutatifs et uniseriels; anneaux dont les ideaux principaux sont idempotents, in Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris, 1982.
  16. B. H. Shafee and S. K. Nauman, On extensions of right symmetric rings without identity, Adv. Pure Math. 4 (2014), no. 12, 665-673. https://doi.org/10.4236/apm.2014.412075
  17. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60 (1974). https://doi.org/10.1090/S0002-9947-1973-0338058-9
  18. J. Wei, Almost Abelian rings, Commun. Math. 21 (2013), no. 1, 15-30.