Existing studies on <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_16_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-prime and weakly <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_16_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-prime submodules have mainly focused on modules over rings, with limited exploration in the setting of semimodules over semirings. In this paper, we extend the definitions and results of Khumprapussorn (Eur J Pure Appl Math 3:730–739, 2018) to <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_16_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-prime subsemimodules of semimodules over commutative semirings. We establish characterizations of weakly <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_16_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-prime subsemimodules and examine their relationships with <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_16_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-prime subsemimodules in both R-semimodule M and the quotient R-semimodule <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_16_Article_IEq12.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$M/N_{(Q)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">/</mo> <msub> <mi>N</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, where N is a partitioning subsemimodule of M. Our results address gaps in previous research and provide new structural insights into prime-like substructures within semimodules over semirings.

On $\alpha $-prime and weakly $\alpha $-prime subsemimodules

期刊论文
发表日期：

2025年11月13日

查看全文

Dipak R. Bonde,
Kunal J. Ingale,
Hemant P. Bendale

摘要

Existing studies on $\alpha $ $α$ -prime and weakly $\alpha $ $α$ -prime submodules have mainly focused on modules over rings, with limited exploration in the setting of semimodules over semirings. In this paper, we extend the definitions and results of Khumprapussorn (Eur J Pure Appl Math 3:730–739, 2018) to $\alpha $ $α$ -prime subsemimodules of semimodules over commutative semirings. We establish characterizations of weakly $\alpha $ $α$ -prime subsemimodules and examine their relationships with $\alpha $ $α$ -prime subsemimodules in both R-semimodule M and the quotient R-semimodule $M/N_{(Q)}$ $M / N_{(Q)}$ , where N is a partitioning subsemimodule of M. Our results address gaps in previous research and provide new structural insights into prime-like substructures within semimodules over semirings.

On \(\alpha \)-prime and weakly \(\alpha \)-prime subsemimodules

摘要