  We introduce the notion of Burch submodules and weakly $\mathfrak m$-full
submodules of modules over local rings and study their properties. One of our
main results shows that Burch submodules satisfy 2-Tor rigid and test property.
We also show that over a local ring $(R,\mathfrak m)$ a submodule $M$ of a
finitely generated $R$-module $X$, such that either $M=\mathfrak m X$ or
$M(\subseteq \mathfrak m X$) is weakly $\mathfrak m$-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 $\mathfrak m$-full). As an application, we give a new
class of rings such that a conjecture of Huneke and Wiegand is affirmative over
them.
