Reflexivity of rings via nilpotent elements

Abdullah Harmanci,Handan Kose,Burcu Ungor,Yosum Kurtulmaz

doi:10.33044/revuma.v61n2a06

Abstract

An ideal I of a ring R is called left N-reflexive if for any a ∈ nil(R) and b ∈ R, aRb ⊆ I implies bRa ⊆ 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.

Highlights

Throughout this paper, all rings are associative with identity
The reversible property of a ring is generalized as: A ring R is said to satisfy the commutativity of nilpotent elements at zero ([2, Definition 2.1]) if ab = 0 for any a, b ∈ nil(R) implies ba = 0; for simplicity, such a ring is called CNZ
We prove that some results of reflexive rings can be extended to the left N-reflexive rings for this general setting

Summary

Introduction

Throughout this paper, all rings are associative with identity. A ring is called reduced if it has no nonzero nilpotent elements. The reversible property of a ring is generalized as: A ring R is said to satisfy the commutativity of nilpotent elements at zero ([2, Definition 2.1]) if ab = 0 for any a, b ∈ nil(R) implies ba = 0; for simplicity, such a ring is called CNZ. In [17], a right ideal I of R is said to be reflexive if aRb ⊆ I implies bRa ⊆ I for any a, b ∈ R. A left ideal I is called idempotent reflexive [11] if aRe ⊆ I implies eRa ⊆ I for a, e2 = e ∈ R. A two sided ideal I of a ring R is called right idempotent reflexive if aRe ⊆ I implies eRa ⊆ I for any a, e2 = e ∈ R.

N-reflexivity of rings

Extensions of N-reflexive rings

Applications

Generalizations and some examples