Journal of Pure and Applied Algebra 28 (198%)211-212 North-Holland Publishing Company 211 AN ADDElNDUM TO “QUADRATIC FORMS OVER POLYNOMIAL EXTENSIONS OF RINGS OF DIMENSION ONE” Journal of Pure and Applied Algebra 24 (1983) 293-302 Raman PARIMALA and Parvin SINCLAIR School of Mathematics, Tata Institute of Fundamental Research, Bombay 400 005, India Communicated by H. Bass Received 8 November 1982 In Theorem 3.3 the condition that disc 4 is extended from R is redundant in view of the following: Proposition. Let R be a reduced commutative Noetherian ring of dimension one in which 2 is invertible and which has finite normalisation. Then the inclusion RGR[X,, . . . , X,,] induces an isomorphism Disc R GDisc R zyxwvutsrqponmlkjihgfedcbaZYXWV [X, , . . . , X,]. To prove the proposition we need the following: Lemma. Let R be any commutative ring il;z which 2 is a non-zero divisor; then zyxwvutsrqp P2W =P2(R[Xl). Proof. Let f=ao+alX+ l *.+ a,X’Ep2(R[X]). Then the equation (ao+alX+-m +a,Xr)2= 1 gives a: = 1 and hence a,o is in p2(R). Let, if possible, i> 0 be the least integer such that ai #I. Then 2aoai == 0 which implies tha.t ai = 0, a contradiction. Thus f = ao. roof of the osition. Let i? be the integral closure of R in its total quotient ring and c be the conductor of R in R. We then have the following commutative diagram of exact sequences: 0022-4049/83/$03.00 iZ:> 1983 North-I-Iolland CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector