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. | Turk J Math (2018) 42: 2081 – 2090 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Existence of maximal ideals in Leavitt path algebras Songül ESİN1,∗,, Müge KANUNİ ER2 , Tüccarbaşı Sok., Kaşe Apt. No: 10A/25, Erenköy, Kadıköy, İstanbul, Turkey 2 Department of Mathematics, Faculty of Arts and Sciences, Düzce University, Konuralp, Düzce, Turkey 1 Received: • Accepted/Published Online: • Final Version: 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 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. Key words: Leavitt path algebras, arbitrary graphs, maximal ideals 1. Introduction and preliminaries It is well known that in a ring with identity, any ideal is contained in a maximal ideal; however, for a nonunital ring, the existence of a maximal ideal is not always guarantied. Also, any maximal ideal is not necessarily a prime ideal in a nonunital case. In this study, for the particular example of a Leavitt path algebra (which is nonunital if the number of vertices of the graph on which it is constructed is infinite), we discuss the existence of maximal ideals and its characterization via the graph properties. The outline of the paper is as follows: we give preliminary definitions in the introduction, and in Section 2, we discuss maximal and prime ideals in a nonunital ring.
