Admissible ideals for k-linear categories

TL;DR

This paper extends the concept of admissible ideals from path algebras to small k-linear categories under certain assumptions, providing new theoretical results and examples.

Contribution

It introduces a generalized framework for admissible ideals in k-linear categories, including relations generated by paths of various lengths, with comprehensive theoretical and example-based insights.

Findings

Generalized admissible ideals to k-linear categories.

Proved analogous results to path algebra cases.

Provided examples and extended discussion on length relations.

Abstract

We generalize the notion of an admissible ideal from path algebras to (small) k-linear categories that satisfy the Krull--Remak--Schmidt--Azumaya assumption. In our treatment we first prove some general results that are analogous to general results for path algebras and admissible ideals. We then cover generalizations of relations generated by paths of length two, which we call point relations, and more general length relations. We conclude the paper with several examples and an appendix containing further discussion on length relations.