On Hereditary Noetherian and Prime Modules over Lavitt Path Algebra: Sink and Infinite Emitter

Risnawita, Risnawita and Gema Hista, Medika and Triyardi Gusti, Ananda (2025) On Hereditary Noetherian and Prime Modules over Lavitt Path Algebra: Sink and Infinite Emitter. Jurnal Matematika UNAND, 14 (4). pp. 390-399. ISSN 2303-291X

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Abstract

Abstract. This study explores the structural and homological characteristics of left modules over Leavitt path algebras, focusing on those generated by specific vertex types in a directed graph, particularly sinks and infinite emitters. The paper examines the module LK(E)u, where u ∈ E0 represents either a sink or an infinite emitter, and determines whether this module exhibits the properties of being hereditary, Noetherian, and prime. Our findings indicate that LK(E)u is indeed hereditary, implying that all its submodules are projective. Additionally, the module satisfies the ascending chain condition, making it Noetherian. However, it fails to qualify as an a-prime module, since there exist nontrivial left ideals I ⊊ LK(E) for which IM ⊊ M, thus does not meet the criteria for primality. These results emphasize how the presence of sinks and infinite emitters significantly affects the module-theoretic behavior of Leavitt path algebras.

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