Absolute Stable Rank and Quadratic Forms over Noncommutative Rings

One o f the ma in aims o f [5] is to obta in bounds for asr(A) for var ious classes of noncommuta t i ve rings A; in part icular , they show that: (i) asr(A) ~< 1 + d whenever A is a module finite a lgebra over a commuta t ive Noe ther ian ring R with d im(maxspec R) = d, and (ii) asr(A) = 1 if A is a semi-local ring (see [5], Theorems 3.1 and 2.4], respectively). The a im o f this note is to show tha t the techniques o f [8] can be easily modif ied to give a simpler p r o o f of these results. Indeed, our p r o o f also works for any f ight Noe ther ian ring, and for any affine PI ring (a ring is called affine if it is finitely generated as an algebra over some central subfield and is PI if it satisfies a po lynomia l identity).