The aim of this article is to provide explicit formulas for the cup product on the Hochschild cohomology of any nonnegatively graded connected algebra and for the cap products on the Hochschild homology of with coefficients in any graded bimodule at the level of the complexes and , resp., where is a minimal projective resolution of the -bimodule , based on the -algebra structure of . We remark that we do not (need to) construct any comparison map between and the Hochschild resolution of , or any lift of the identity of . The main tools we use come from torsion theory of -algebras and of their -bimodules.