REFLEXIVITY OF RINGS VIA NILPOTENT ELEMENTS

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Harmanci A., KÖSE H., KURTULMAZ Y., ÜNGÖR B.

REVISTA DE LA UNION MATEMATICA ARGENTINA, vol.61, no.2, pp.277-290, 2020 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 61 Issue: 2
  • Publication Date: 2020
  • Doi Number: 10.33044/revuma.v61n2a06
  • Journal Name: REVISTA DE LA UNION MATEMATICA ARGENTINA
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, Directory of Open Access Journals, DIALNET
  • Page Numbers: pp.277-290
  • Keywords: Reflexive ring, left N-reflexive ring, left N-right idempotent reflexive ring, quasi-Armendariz ring, nilpotent element, PROPERTY, MODULES
  • Ankara University Affiliated: Yes

Abstract

An ideal I of a ring R is called left N-reflexive if for any a 2 nil(R) and b is an element of R, aRb subset of I implies bRa subset of I, where nil(R) is the set of all nilpotent elements of R. The ring R is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive rings, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal I of a ring R, R/I is left N-reflexive. If an ideal I of a ring R is reduced as a ring without identity and R/I is left N-reflexive, then R is left N-reflexive. If R is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in R[x] are nilpotent in R, it is proved that R is left N-reflexive if and only if R[x] is left N-reflexive. We show that the concept of left N-reflexivity is weaker than that of reflexivity and stronger than that of right idempotent reflexivity.