Let \(\mathcal C\) be the category whose objects consist of \(R\)-algebras (possibly non-commutative) over a possibly non-commutative ring and whose opposite morphisms consist of direct sums of bi-flat epimorphisms \(A\to B_{i}\). A map \(A\to B\) of \(R\)-algebras is a bi-flat epimorphism if every \(A\)-algebra map \(g: B\to C\) is unique. Bi-flat epimorphisms are viewed dually as localizations. An \(R\)-algebra \(A\) in \(\mathcal C\), denoted \(Geo(A)\) as an element of \(\mathcal C\), acquires via such localizations a Grothendieck topology and the representation functor is a sheaf for this Grothendieck topology. An analogue of affine scheme is thus introduced in the non-commutative case.