Vanishing of (co)homology of Burch and related submodules (Q6100870)

As authors mentioned in the abstract, they introduce the notion of Burch submodules and weakly \(\mathfrak{m}\)-full submodules of modules over a local ring \((R,\mathfrak{m})\) and study their properties. They show that Burch submodules satisfy 2-Tor rigid and test properties. They also show that over a local ring \((R,\mathfrak{m})\), a submodule \(M\) of a finitely generated \(R\)-module \(X\), such that either \(M=\mathfrak{m}M\) or \(M(\subseteq\mathfrak{m}M\)) 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, they give some new class of modules and a new class of rings such that a conjecture of Huneke and Wiegand is affirmative for them. Recall that an \(R\)-module \(M\) is called \textit{\(n\)-Tor rigid} if \(\mathrm{Tor}_i^R(M,N)=0\) for \(i=t+1,\ldots,t+n\) implies \(\mathrm{Tor}_j^R(M,N)=0\) for all \(j>t\). And \(M\) is called \textit{a test module} (for projectivity), if for all \(R\)-module \(N\), \(\mathrm{Tor}_i^R(M,N)=0\) for all sufficiently large \(i >0\) implies \(\mathrm{pd}_RN <\infty\).