the first adequate definition of "k" of a ring was made by hyman bass and stephen schanuel. in topological "k"-theory, "k" is defined using vector bundles on a suspension of the space. all such vector bundles come from the clutching construction, where two trivial vector bundles on two halves of a space are glued along a common strip of the space. this gluing data is expressed using the general linear group, but elements of that group coming from elementary matrices (matrices corresponding to elementary row or column operations) define equivalent gluings. motivated by this, the bass–schanuel definition of "k" of a ring "r" is , where "gl"("r") is the infinite general linear group (the union of all "gl"("r")) and "e"("r") is the subgroup of elementary matrices. they also provided a definition of "k" of a homomorphism of rings and proved that "k" and "k" could be fit together into an exact sequence similar to the relative homology exact sequence.