A generalization of the theory of standardly stratified algebras I: Standardly stratified ringoids

TL;DR

This paper generalizes the concept of standardly stratified algebras to rings with enough idempotents, introducing new classes like ideally standardly stratified and ideally quasi-hereditary rings, and explores their properties and differences from classical cases.

Contribution

It extends classical theory to a broader class of rings, establishing new classes and analyzing their relationships, especially between quasi-hereditary and ideally quasi-hereditary rings.

Findings

Many classical results are generalized to rings without unity.

New classes of rings such as ideally standardly stratified are introduced.

The equivalence between quasi-hereditary and ideally quasi-hereditary rings no longer holds in this setting.

Abstract

We extend the classical notion of standardly stratified $k$-algebra (stated for finite dimensional $k$-algebras) to the more general class of rings, possibly without $1,$ with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting this is no longer true. To develop the theory, we use the well known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).