ON HEREDITARY NOETHERIAN AND PRIME MODULES OVER LEAVITT PATH ALGEBRA: SINK AND INFINITE EMITTER - Risnawita, Risnawita; GEMA HISTA MEDIKA; ANANDA, TRIYARDI GUSTI (Jurnal Matematika UNAND, 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 \( u \in E^0 \) 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 \( I \subsetneq L_K(E) \) for which \( IM \subsetneq M \), 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.