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.
Leavitt path algebras, arbitrary graphs, maximal ideals
ESİN, SONGÜL and ER, MÜGE KANUNİ
"Existence of maximal ideals in Leavitt path algebras,"
Turkish Journal of Mathematics: Vol. 42:
5, Article 1.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss5/1