Thenecessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed.
📌 3 Key Takeaways
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Objective & Hypothesis:Let E be an arbitrary directed graph and let L be the Leavitt path algebra of the graph E over a field K.The necessary and sufficient conditions are given to assure the existence of a maximal ideal in L and also the necessaryand sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given.
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Key Findings & Data:It is shownthat if a maximal ideal M of L is nongraded, then the largest graded ideal in M, namely gr(M) , is also maximalamong the graded ideals of L.
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Conclusion & Impact:Thenecessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
Let E be an arbitrary directed graph and let L be the Leavitt path algebra of the graph E over a field K.The necessary and sufficient conditions are given to assure the existence of a maximal ideal in L and also the necessaryand sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given. It is shownthat if a maximal ideal M of L is nongraded, then the largest graded ideal in M, namely gr(M) , is also maximalamong the graded ideals of L. Moreover, if L has a unique maximal ideal M, then M must be a graded ideal. Thenecessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed.
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Cite This Article
ESİN, S., ER, M. K. (2022). Existence of maximal ideals in Leavitt path algebras. Turkish Journal of Mathematics, Cilt: 42 - Sayı: 5, 2081–2090. https://ajindex.com/articles/607759
Abrams G, Ara P, Siles Molina M. Leavitt Path Algebras. Lecture Notes in Mathematics. London, UK: Springer- Verlag, 2017.
Abrams G, Aranda Pino G. The Leavitt path algebra of a graph. J Algebra 2005; 293: 319-334.
Abrams G, Aranda Pino G. The Leavitt path algebra of arbitrary graphs. Houston J Math 2008; 34: 423-442.
Anderson FW, Fuller KR. Rings and Categories of Modules. Graduate Texts in Mathematics, Vol 13, Berlin, Germany: Springer-Verlag, 1974.
Ara P, Moreno MA, Pardo E. Nonstable K-theory for graph algebras. Algebr Represent Th 2007; 10: 157-178.
Esin S, Kanuni M, Rangaswamy KM. On the intersections of two-sided ideals of Leavitt path algebras. J Pure Appl Algebra 2017; 221: 632-644. Rangaswamy KM. The theory of prime ideals of Leavitt path algebras over arbitrary graphs. J Algebra 2013; 375: 73-96. [8] arbitrary graphs. Commun Algebra 2014; 42: 2859-2868. Tomforde T. Uniqueness theorems and ideal structure of Leavitt path algebras. J Algebra 2007; 318: 270-299.