On intersections of two-sided ideals of Leavitt path algebras

Abstract

Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. It is shown that every ideal of $L$ is an intersection of primitive/prime ideals in $L$ if and only if the graph $E$ satisfies Condition (K). Uniqueness theorems in representing an ideal of $L$ as an irredundant intersection and also as an irredundant product of finitely many prime ideals are established. Leavitt path algebras containing only finitely many prime ideals and those in which every ideal is prime are described. Powers of a single ideal $I$ are considered and it is shown that the intersection ${\displaystyle\bigcap\limits_{n=1}^{\infty}}I^{n}$ is the largest graded ideal of $L$ contained in $I$. This leads to an analogue of Krull's theorem for Leavitt path algebras.