An account on certain annihilating properties in Leavitt path algebras.

Abstract

We present that for any graph E and for a commutative unital ring R, the nil ideals of the Leavitt path algebra LR(E) depend solely only the nil ideals of the ring R. A connection between the Jacobson radical of LR(E) and the Jacobson radical of R is obtained. Further we show that for a nil ideal I of a Leavitt path algebra LR(E) the ideal M2(I) is also nil. Thus obtaining that Leavitt path algebras over arbitrary graphs satisfy the K¨oethe’s conjecture.