We study <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> properties of rings in a more general setting by introducing the concept of t-<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> rings, defined via a ring tripotent, and investigating their properties. We prove that every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> ring is a t-<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> ring, but the converse fails, as we demonstrate by a counterexample. Additional examples and counterexamples are provided to illustrate these results. Moreover, we examine t-<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> properties of rings relative to a ring endomorphism <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. Some results on reversible rings, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> rings and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-skew <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{CNZ}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>CNZ</mtext> </math></EquationSource> </InlineEquation> rings are extended and unified (see [<CitationRef CitationID="CR1">1</CitationRef>, <CitationRef CitationID="CR2">2</CitationRef>] and [<CitationRef CitationID="CR4">4</CitationRef>]).

Zero commutativity of nilpotent elements via tripotents

期刊论文
发表日期：

2025年12月1日

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Liang Zhao,
Haiyun Wu

摘要

We study $\textrm{CNZ}$ $CNZ$ properties of rings in a more general setting by introducing the concept of t- $\textrm{CNZ}$ $CNZ$ rings, defined via a ring tripotent, and investigating their properties. We prove that every $\textrm{CNZ}$ $CNZ$ ring is a t- $\textrm{CNZ}$ $CNZ$ ring, but the converse fails, as we demonstrate by a counterexample. Additional examples and counterexamples are provided to illustrate these results. Moreover, we examine t- $\textrm{CNZ}$ $CNZ$ properties of rings relative to a ring endomorphism $\alpha $ $α$ . Some results on reversible rings, $\textrm{CNZ}$ $CNZ$ rings and $\alpha $ $α$ -skew $\textrm{CNZ}$ $CNZ$ rings are extended and unified (see [1, 2] and [4]).