On Gerko's Strongly Tor-independent Modules

Gerko proves that if an artinian local ring $(R,{\mathfrak{m}}_R)$ possesses a sequence of strongly Tor-independent modules of length $n$, then ${\mathfrak{m}}_R^n\mathrm{\ne }0$. This generalizes readily to Cohen-Macaulay rings. We present a version of this result for non-Cohen-Macaulay rings.