Characteristics of Rota-Baxter Algebras

TL;DR

This paper investigates the invariant properties of Rota-Baxter algebras, introduces the ascent set to classify their characteristics, and explores their structure and prime characteristics across different base rings.

Contribution

It introduces the ascent set invariant, classifies Rota-Baxter characteristics in the homogeneous case, and relates general characteristics to the homogeneous case via initial ideals.

Findings

Classification of Rota-Baxter characteristics in the homogeneous case

Relation of general characteristics to homogeneous case through initial ideals

Determination of prime characteristics of Rota-Baxter rings

Abstract

The characteristic is a simple yet important invariant of an algebra. In this paper, we study the characteristic of a Rota-Baxter algebra, called the Rota-Baxter characteristic. We introduce an invariant, called the ascent set, of a Rota-Baxter characteristic. By studying its properties, we classify Rota-Baxter characteristics in the homogenous case and relate Rota-Baxter characteristics in general to the homogeneous case through initial ideals. We show that the Rota-Baxter quotients of Rota-Baxter characteristics have the same underlying sets as those in the homogeneous case. We also give a more detailed study of Rota-Baxter characteristics with special base rings. In particular, we determine the prime characteristics of Rota-Baxter rings.