Simpleness of Leavitt Path Algebras with Coefficients in a Commutative Semiring

TL;DR

This paper characterizes ideal- and congruence-simpleness of Leavitt path algebras over commutative semirings, extending known results from fields and rings to semifields and more general semirings.

Contribution

It provides complete characterizations of ideal- and congruence-simpleness for Leavitt path algebras over commutative semirings, generalizing previous results from fields and rings.

Findings

Characterization of ideal-simple Leavitt path algebras over semifields.

Complete description of congruence-simple Leavitt path algebras over row-finite graphs.

Extension of known algebraic properties from fields to semirings.

Abstract

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, as well as establish some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a semifield S that extends the well-known characterizations when the ground semiring S is a field. Also, extending the well-known characterizations when S is a field or commutative ring, we present a complete characterization of congruence-simple Leavitt path algebras over row-finite graphs with coefficients in a commutative semiring S.