Several different constructions of a structure sheaf over noetherian noncommutative rings have been made in the recent past; however the problem of functoriality remains open in the general case. Here we present a new approach to the problem. Since the two-sided structure of the ring is fundamental in finding a solution of the above problem it seems natural to evaluate the sheaf functors on (normalizing) bimod-ules. In this context, biradicals arise as a useful tool as well as classical localization at prime ideals and cliques. The main results are valid for noetherian rings satisfying the strong second layer condition in which every clique is classically localizable. This paper can be considered an extension of [5], and develops its results outside of FBN rings.