Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs

The first goal of the present paper is to study class groups edge rings complete multipartite graphs, denoted by k[Kr1,…,rn], where 1≤r1≤⋯≤rn. More concretely, we prove that group k[Kr1,…,rn] isomorphic Zn if n=3 with r1≥2 or n≥4, while it turns out excluded cases can be deduced into Hibi rings. second investigate special divisorial ideals called conic ideals. We describe for certain Kr1,…,rn including all Gorenstein. Finally, give a non-commutative crepant resolution (NCCR) in case