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<p>A method of Rosenmann and Rosset for computing module type of fc-localization of free algebras is used to determine module type of quotient rings of free algebras. The now widely studied classical Leavitt algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{L}_{K}(1,n)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="double-struck">L</mi> </mrow> <mrow> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> over a field <i>K</i> is seen to be a ring of right quotients of the unital free associative algebra of rank <i>n</i> with respect to the perfect Gabriel ideal topology of a 1-codimensional ideal, i.e., by a nonunital free associative algebra, providing a conceptual, variable-free description of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{L}_{K}(1,n)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="double-struck">L</mi> </mrow> <mrow> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>. This result puts Leavitt (path) algebras on the frontier of important research areas in localization theory, quiver algebras, graph operator algebras, and in the study of free ideal rings and their automorphism groups.</p>

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Module types of localizations, with applications to Leavitt path algebras

  • 期刊论文

  • 发表日期：

    2025年9月4日

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  • Pham Ngoc Ánh,
  • Michael F. Siddoway

摘要

A method of Rosenmann and Rosset for computing module type of fc-localization of free algebras is used to determine module type of quotient rings of free algebras. The now widely studied classical Leavitt algebra \(\mathbb{L}_{K}(1,n)\) L K ( 1 , n ) over a field K is seen to be a ring of right quotients of the unital free associative algebra of rank n with respect to the perfect Gabriel ideal topology of a 1-codimensional ideal, i.e., by a nonunital free associative algebra, providing a conceptual, variable-free description of \(\mathbb{L}_{K}(1,n)\) L K ( 1 , n ) . This result puts Leavitt (path) algebras on the frontier of important research areas in localization theory, quiver algebras, graph operator algebras, and in the study of free ideal rings and their automorphism groups.