Flat modules and semiperfect rings

Abstract

Let A be a ring, E _A and _A M be right and left A-modules, respectively, and let E × M be the cartesian product of these modules. The tensor product E⊖ _A M is the Abelian group F/H, where F is the free Z-module with basis indexed by E × M, and H is the subgroup of F generated by all elements of the form

$$ (x + u,y) - (x,y) - (u,y),(x,y + v) - (x,y) - (x,v),(xa,y) - (x,ay), $$

where x, u ∈ E, y, v ∈ M, and a ∈ A.(We write E ⊖ M instead of E ⊖ _A M if there is no doubt about A.) The image of (x, y) under a natural map E × M → E ⊖ M is denoted by x ⊖ y.