On Endomorphism Rings of Leavitt Path Algebras

Let E be an arbitrary graph, K be any field and A be the endomorphism ring of L := L-K(E) considered as a right L-module. Among the other results, we prove that: (1) if A is a von Neumann regular ring, then A is dependent if and only if for any two paths in L satisfying some conditions are initial of each other, (2) if A is dependent then L-K(E) is morphic, (3) L is morphic and von Neumann regular if and only if L is semisimple and every homogeneous component is artinian.