Characterizing Rickart and Baer ultragraph Leavitt path algebras

TL;DR

This paper provides a comprehensive characterization of ultragraph Leavitt path algebras that are Rickart, Baer, and their variants, extending previous work from fields to semi-simple commutative unital rings.

Contribution

It generalizes existing characterizations of Leavitt path algebras over fields to the broader class of ultragraph Leavitt path algebras over semi-simple commutative unital rings.

Findings

Characterization of Rickart ultragraph Leavitt path algebras

Characterization of Baer ultragraph Leavitt path algebras

Extension of previous work to semi-simple rings

Abstract

We characterize ultragraph Leavitt path algebras that are Rickart, locally Rickart, graded Rickart, and graded Rickart *-rings. We also characterize ultragraph Leavitt path algebras that are Baer, locally Baer, graded Baer, Baer *-rings, and combinations of these. These characterizations build on and generalize the work of Hazrat and Vas on Leavitt path algebras over fields to ultragraph Leavitt path algebras over semi-simple commutative unital rings.