  Given a finite module $M$ over a Noetherian local ring $(R, \m)$, we
introduce the concept of $j$-stretched ideals on $M$. Thanks to a crucial
specialization lemma, we show that this notion greatly generalizes (to
arbitrary ideals, and with respect to modules) the classical definition of
stretched $\m$-primary ideals of Sally and Rossi-Valla, as well as the notion
of minimal and almost minimal $j$-multiplicity given recently by Polini-Xie.
For $j$-stretched ideals $I$ on a Cohen-Macaulay module $M$, we show that ${\rm
gr}_I(M)$ is Cohen-Macaulay if and only if two classical invariants of $I$, the
reduction number and the index of nilpotency, are equal. Moreover, for the same
class of ideals, we provide a generalized version of Sally's conjecture
(proving the almost Cohen-Macaulayness of associated graded rings). Our work
unifies the approaches of Rossi-Valla and Polini-Xie and generalizes
simultaneously results on the (almost) Cohen-Macaulayness %and almost
Cohen-Macaulayness of associated graded modules by several authors, including
Sally, Rossi-Valla, Wang, Elias, Rossi, Corso-Polini-Vaz Pinto, Huckaba and
Polini-Xie.
