<p>In this paper, we introduce strongly quasi primary submodules. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R\ \)</EquationSource> </InlineEquation>be a commutative ring with nonzero identity and <i>M</i> be a unital <i>R</i>-module and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(N\ \)</EquationSource> </InlineEquation>a proper submodule of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(M.\ \)</EquationSource> </InlineEquation>Then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(N\ \)</EquationSource> </InlineEquation>is called strongly quasi primary if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(rm\in N\ \)</EquationSource> </InlineEquation>for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(r\in R\ \)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(m\in M\)</EquationSource> </InlineEquation> implies either <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(r^{2}\in (N:M)\ \)</EquationSource> </InlineEquation>or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(m\in rad(N).\ \)</EquationSource> </InlineEquation>In addition to give many properties and examples of strongly quasi primary submodules, we use them to characterize some special modules such as von Neumann regular modules and divided modules. Also, we investigate the strongly quasi primary ideals/submodules of amalgamations and give the conditions under which amalgamated duplication of a module <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(M\bowtie I\)</EquationSource> </InlineEquation> is a divided module.</p>

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On Strongly Quasi Primary Submodules

  • Gülşen Ulucak,
  • Suat Koç,
  • Ünsal Tekir

摘要

In this paper, we introduce strongly quasi primary submodules. Let \(R\ \) be a commutative ring with nonzero identity and M be a unital R-module and \(N\ \) a proper submodule of \(M.\ \) Then \(N\ \) is called strongly quasi primary if \(rm\in N\ \) for \(r\in R\ \) and \(m\in M\) implies either \(r^{2}\in (N:M)\ \) or \(m\in rad(N).\ \) In addition to give many properties and examples of strongly quasi primary submodules, we use them to characterize some special modules such as von Neumann regular modules and divided modules. Also, we investigate the strongly quasi primary ideals/submodules of amalgamations and give the conditions under which amalgamated duplication of a module \(M\bowtie I\) is a divided module.