Abstract We study the structure and compute stable rank of $C^{*}$ -algebras finite higher-rank graphs. completely determine -algebra when $k$ -graph either contains no cycle with an entrance or is cofinal. also exactly which finite, locally convex -graphs yield unital stably -algebras. give several examples to illustrate our results.

In this paper, we study the class of rings that satisfy internal direct sum cancellation with respect to their 1-sided ideals. These are known to be precisely the rings in which regular elements are unitregular. Further characterizations for such “IC rings” are given, in terms of suitable versions of stable range conditions, and unique generator properties of idempotent generated right ideals. ...

A technique of minimal free resolutions of Stanley–Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial (d − 1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d − 1 is d-connected.

This paper classifies the Grothendieck rings of complex fusion categories multiplicity one up to rank six. Among 72 possible rings, $25$ ones are filtered out by using categorification criteria. Each remaining 47 admits a unitary categorification. We found 6 new categorified applying localization approach Pentagon Equation.