Simple Cuntz-Pimsner rings

TL;DR

This paper establishes criteria for simplicity and ideal structure in relative Cuntz-Pimsner rings, introduces conditions (L) and (K), and proves a Cuntz-Krieger uniqueness theorem, advancing understanding of their algebraic properties.

Contribution

It provides necessary and sufficient conditions for ideal properties, introduces conditions (L) and (K), and proves a Cuntz-Krieger uniqueness theorem for relative Cuntz-Pimsner rings.

Findings

Criteria for when every non-zero ideal contains a non-zero graded ideal.

Conditions under which a relative Cuntz-Pimsner ring is simple.

A Cuntz-Krieger uniqueness theorem for these rings.

Abstract

Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner ring is graded, are given. A "Cuntz-Krieger uniqueness theorem" for relative Cuntz-Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz-Pimsner rings are introduced.