Sally modules and reduction numbers of ideals - Ghezzi, L.; Goto, S.; Hong, J.; Vasconcelos, W. V. (Nagoya Mathematical Journal, 2017)
We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) to develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen–Macaulay rings to more general rings; (ii) to use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen–Macaulayness of the special fiber ring; (iii) to extend some of the results of (i) to two-dimensional Buchsbaum rings. Along the way, we provide an explicit realization of the S2S_{2} -fication of arbitrary Buchsbaum rings.
Keywords: 13D40; 13H15; Primary 13H10; Secondary 13A30; dept.mat

Generalization of bi-canonical degrees - Brennan, J.; Ghezzi, L.; Hong, J.; Vasconcelos, W. V. (São Paulo Journal of Mathematical Sciences, 2023)
We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $$ω$$. Attached to each such ring R, when $$ω$$is an ideal, there are integers–the type of R, the reduction number of $$ω$$–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 $$ω$$is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.
Keywords: 13A99; Bi-canonical degree; Canonical module; Primary 13H15; Secondary 13H10; dept.mat; peer-reviewed