Existence of maximal ideals in Leavitt path algebras.

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 necessary and sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given. It is shown that if a maximal ideal M of L is nongraded, then the largest graded ideal in M, namely gr(M), is also maximal among the graded ideals of L. Moreover, if L has a unique maximal ideal M, then M must be a graded ideal. The necessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed. [ABSTRACT FROM AUTHOR]