Tor-pairs: products and approximations

Abstract: 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 $(\mathcal T,\mathcal S)$, relating this property with the relative $\mathcal T$-Mittag-Leffler dimension of modules in $\mathcal S$. 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.