Let N be a near-ring with identity 1 and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_785_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{n}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the near-ring of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_785_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$n\times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrices over N for an arbitrary natural number n. The main aims of this paper are to introduce and study the notions of quasi-ideal and equiprime quasi-ideal of a near-ring N. It is shown that there is a one-to-one correspondence between the set of quasi-ideals of N and the set of full quasi-ideals of matrix near-ring <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_785_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{n}(N).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Furthermore, there is a one-to-one correspondence between the set of equiprime quasi-ideals of N and the set of equiprime quasi-ideals of matrix near-ring <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_785_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$M_{n}(N).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Some generalizations of ideals in near-rings

期刊论文
发表日期：

2025年4月18日

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Ahmed Y. Abdelwanis,
Shakir Ali,
Vaishali Varshney,
Olfat A. Elmenofy

摘要

Let N be a near-ring with identity 1 and $M_{n}(N)$ $M_{n} (N)$ be the near-ring of $n\times n$ $n \times n$ matrices over N for an arbitrary natural number n. The main aims of this paper are to introduce and study the notions of quasi-ideal and equiprime quasi-ideal of a near-ring N. It is shown that there is a one-to-one correspondence between the set of quasi-ideals of N and the set of full quasi-ideals of matrix near-ring $M_{n}(N).$ $M_{n} (N) .$ Furthermore, there is a one-to-one correspondence between the set of equiprime quasi-ideals of N and the set of equiprime quasi-ideals of matrix near-ring $M_{n}(N).$ $M_{n} (N) .$