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 arXiv:1701.05592 and arXiv:1711.09480 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.