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<p>This paper introduces the notion of uniformly <i>S</i>-pseudo-injective (<i>u</i>-<i>S</i>-pseudo-injective) modules as a generalization of <i>u</i>-<i>S</i>-injective modules. Let <i>R</i> be a ring and <i>S</i> a multiplicative subset of <i>R</i>. An <i>R</i>-module <i>E</i> is said to be <i>u</i>-<i>S</i>-pseudo-injective if for any submodule <i>K</i> of <i>E</i>, there is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s\in S\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> such that for any <i>u</i>-<i>S</i>-monomorphism <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f:K\rightarrow E\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation>, <i>sf</i> can be extended to an endomorphism <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(g:E\rightarrow E\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation>. Several properties of this notion are studied. For example, we show that an <i>R</i>-module <i>M</i> is <i>u</i>-<i>S</i>-quasi-injective if and only if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(M\oplus M\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>⊕</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> is <i>u</i>-<i>S</i>-pseudo-injective. Two classes of rings related to the class of <i>QI</i>-rings are introduced and characterized.</p>

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Uniformly S-pseudo-injective modules

  • 期刊论文

  • 发表日期：

    2026年2月17日

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  • Mohammad Adarbeh,
  • Mohammad Saleh

摘要

This paper introduces the notion of uniformly S-pseudo-injective (u-S-pseudo-injective) modules as a generalization of u-S-injective modules. Let R be a ring and S a multiplicative subset of R. An R-module E is said to be u-S-pseudo-injective if for any submodule K of E, there is \(s\in S\) s ∈ S such that for any u-S-monomorphism \(f:K\rightarrow E\) f : K → E , sf can be extended to an endomorphism \(g:E\rightarrow E\) g : E → E . Several properties of this notion are studied. For example, we show that an R-module M is u-S-quasi-injective if and only if \(M\oplus M\) M ⊕ M is u-S-pseudo-injective. Two classes of rings related to the class of QI-rings are introduced and characterized.