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<p>Let <i>R</i> be a ring and <i>M</i> be a left <i>R</i>-module. We call <i>M ps-lifting</i> every submodule <i>N</i> of <i>M</i> containsa direct summand <i>X</i> of <i>M</i> such that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{N}{X}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi>N</mi> <mi>X</mi> </mfrac> </math></EquationSource> </InlineEquation> is projective semisimple. In this paper, we provide some properties of these modules. It is shown that: (1) if a projective module is ps-lifting, then it is hereditary; (2) for a ring <i>R,</i> every left <i>R</i>-module is ps-lifting if and only if every <i>R</i>-module is a direct sum of an injective module and a projective semisimple module; (3) <sub><i>R</i></sub><i>R</i> is ps-lifting if and only if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\frac{R}{\text{Soc}\left(R\right)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi>R</mi> <mrow> <mtext>Soc</mtext> <mfenced close=")" open="("> <mi>R</mi> </mfenced> </mrow> </mfrac> </math></EquationSource> </InlineEquation> is semisimple and <i>R</i> is hereditary.</p>

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PS-Lifting Modules

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • Engin Kaynar

摘要

Let R be a ring and M be a left R-module. We call M ps-lifting every submodule N of M containsa direct summand X of M such that \(\frac{N}{X}\) N X is projective semisimple. In this paper, we provide some properties of these modules. It is shown that: (1) if a projective module is ps-lifting, then it is hereditary; (2) for a ring R, every left R-module is ps-lifting if and only if every R-module is a direct sum of an injective module and a projective semisimple module; (3) RR is ps-lifting if and only if \(\frac{R}{\text{Soc}\left(R\right)}\) R Soc R is semisimple and R is hereditary.