Generalized stretched ideals and Sally Conjecture

We introduce the concept of $j$-stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched $\mathfrak{m}$-primary ideals of Sally and Rossi-Valla, as well as the concept of ideals of minimal and almost minimal $j$-multiplicity introduced by Polini-Xie. One of our main theorems states that, for a $j$-stretched ideal, the associated graded ring is Cohen-Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem, presenting numerical conditions which ensure the almost Cohen-Macaulayness of the associated graded ring of a $j$-stretched ideal, provides a generalized version of Sally's conjecture. This work, which also holds for modules, unifies the approaches of Rossi-Valla and Polini-Xie and generalizes simultaneously results on the Cohen-Macaulayness or almost Cohen-Macaulayness of the associated graded module by several authors, including Sally, Rossi-Valla, Wang, Elias, Corso-Polini-Vaz Pinto, Huckaba, Marley and Polini-Xie.