An elementary description of K₁(R) without elementary matrices

Abstract

Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R) → GL(n+1,R) (which add a unit vector as new last row and column) yields, by definition, the stable linear group GL(R); the same result is obtained, up to isomorphism, when using the “opposite” inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic K-group K₁(R) = GL(R)/E(R) of R, giving an elementary description that does not involve elementary matrices explicitly.