Admissible ideals for k-linear categories

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.