ON HEREDITARY NOETHERIAN AND PRIME MODULES OVER LEAVITT PATH ALGEBRA: SINK AND INFINITE EMITTER - Risnawita, Risnawita; GEMA HISTA MEDIKA; ANANDA, TRIYARDI GUSTI (Departemen Matematika dan Sains Data FMIPA Universitas Andalas Padang, 2025)
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 \( L_K(E)u \), where \( u E^0 \) 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 \( L_K(E)u \) 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 \( I L_K(E) \) 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.