Tor-pairs: products and approximations

Recently the author has studied rings for which products of flat modules have finite flat dimension. In this paper we extend the theory to characterize when products of modules in $\mathcal{T}$ have finite $\mathcal{T}$-projective dimension, where $\mathcal{T}$ is the left hand class of a Tor-pair $(\mathcal T,\mathcal S)$,relatingthispropertywiththerelative$\mathcal T$-Mittag-Lefflerdimensionofmodulesin$\mathcal S$. We apply these results to study the existence of approximations by modules in $\mathcal{T}$. In order to do this, we give short proofs of the well known results that a deconstructible class is precovering and that a deconstructible class closed under products is preenveloping.