Existence of maximal ideals in Leavitt path algebras

Abstract

Let E be an arbitrary directed graph and let L be the Leavitt path algebra of the graph E over a field K.The necessary and sufficient conditions are given to assure the existence of a maximal ideal in L and also the necessaryand sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given. It is shownthat if a maximal ideal M of L is nongraded, then the largest graded ideal in M, namely gr(M) , is also maximalamong the graded ideals of L. Moreover, if L has a unique maximal ideal M, then M must be a graded ideal. Thenecessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed.