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<p>In this paper, we discuss some properties on Lucas modules. In details, we show that direct and inverse limits of Lucas modules are Lucas modules, and every <i>R</i>-module has a Lucas envelope and a Lucas cover. Moreover, some properties of direct and inverse limits of Lucas modules and some constructions and the unique mapping properties of Lucas envelopes and Lucas covers are investigated. As an application, we show that every <i>R</i>-module has a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>-cover with the unique mapping property, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> is the class of all vector spaces over <i>Q</i>, the quotient field of an integral domain <i>R</i>.</p>

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Some remarks on Lucas modules

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Xiaolei Zhang,
  • Guocheng Dai,
  • Wei Qi

摘要

In this paper, we discuss some properties on Lucas modules. In details, we show that direct and inverse limits of Lucas modules are Lucas modules, and every R-module has a Lucas envelope and a Lucas cover. Moreover, some properties of direct and inverse limits of Lucas modules and some constructions and the unique mapping properties of Lucas envelopes and Lucas covers are investigated. As an application, we show that every R-module has a \(\mathcal {V}\) V -cover with the unique mapping property, where \(\mathcal {V}\) V is the class of all vector spaces over Q, the quotient field of an integral domain R.