ON LOCALLY UNIT REGULARITY CONDITIONS FOR ARBITRARY LEAVITT PATH ALGEBRAS

Let Γ be a graph, K be any field and S be the endomorphism ring of L := LK(Γ) considered as a right L-module. In this paper, we give defination of the left locally unit regular ring. We show that (1) if S is locally unit regular, then L is locally unit regular, (2) if L is morphic and image projective then S is left morphic, (3) S is a directly finite ring then L is directly finite, (4) if S is an exchange ring then L is directly finite and if L is a directly finite ring then L is an exchange ring.