Generalization of bi-canonical degrees

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $$\omega$$. Attached to each such ring R, when $$\omega$$is an ideal, there are integers–the type of R, the reduction number of $$\omega$$–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where $$\omega$$is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.