Basic One-Sided Ideals of Leavitt Path Algebras over Commutative Rings

Abstract

AbstractIn this article, basic left (right) ideals of Leavitt Path Algebra over a commutative unital ring are studied. We give conditions under which a basic left (right) ideal generated by a vertex is a minimal basic left (right) ideal. It is further shown that if R has no non-zero nilpotent elements, then every minimal basic left ideal \(L_R(E)x\) of the Leavitt path algebra \(L_R(E)\) contains a vertex. Among other techniques, the proof depends on the fact that a Leavitt Path Algebra over a commutative unital ring R is non-degenerate if and only if R has no non-zero nilpotent elements (equivalently, R is a (commutative) semiprime ring).KeywordsLeavitt path algebraBasic left idealMinimal basic left ideal