Vanishing of (co)homology of Burch and related submodules

We introduce the notion of Burch submodules and weakly m-full submodules of modules over a local ring (R,m) and study their properties. One of our main results shows that Burch submodules satisfy 2-Tor rigid and test properties. We also show that over a local ring (R,m), a submodule M of a finitely generated R-module X, such that either M=mX or M(⊆mX) is weakly m-full in X, is 1-Tor rigid, and a test module provided that X is faithful (and X∕M has finite length when M is weakly m-full). As an application, we give some new class of modules and a new class of rings such that a conjecture of Huneke and Wiegand is affirmative for them.