On nil ideals of Leavitt path algebras over commutative rings

Abstract

We show in this paper that for any graph E and for a commutative unital ring R, the nil ideals of the Leavitt path algebra L R (E) depend solely on the nil ideals of the ring R. A connection between the Jacobson radical of L R (E) and the Jacobson radical of R is obtained. We also prove that for a nil ideal I of a Leavitt path algebra L R (E) the ideal M 2 (I) is also nil, thus obtaining that Leavitt path algebras over arbitrary graphs satisfy the Köethe's conjecture.