Primeness of Simple Modules over Path Algebras and Leavitt Path Algebras

Abstract

Let K be a field and E be a directed graph, called quiver in the following, and let A = KE be the path algebra that corresponds to E with coefficients in K. An A-module M is a c-prime module in the sense that rm = 0 for one m in M and r in A implies that either r annihilates all M or m = 0. In this paper, we prove that for any acyclic graph E, an A-module M is c-prime if and only if it is simple. The primeness of simple modules over Leavitt path algebras is also discussed. We prove that some classes of simple modules over Leavitt path algebras, are not c-prime modules.