PRIMENESS OF SIMPLE MODULES OVER PATH ALGEBRAS AND LEAVITT PATH ALGEBRAS - Risnawita
The concept of primeness in algebraic structures was initially developed through the ideal structure of the ring. The concept of a prime ideal was introduced as the concept of a prime ring. Suppose thatM is a left module over the ring R (written R - moduleM). A proper submodule N ofM is said to be prime if rRm = 0 with r in R, and m ? M implies m ? N or rM ? N. An R-module M is a c-prime module in the sense that rm = 0 for one m ? M and r ? R implies that either r annihilates allM orm = 0. The aim of this study was to explore the notion of c-prime modules in the setting of path algebras and Leavitt path algebras. Let K be a field and E be a directed graph, and let A = KE be the path algebra that correspondence to E with coefficients in K. 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. We also discussed about primeness of simple modules over Leavitt path algebra. We prove that for some classes of simple modules over Leavitt path algebra, they are not c-prime modules. Moreover, we have also provided the necessary and sufficient conditions for the graph E so that there is a simple moduleM which is c-prime. We also discussed about primeness of graded simple modules over Leavitt path algebra.