Estes, Dennis R.; Guralnick, Robert M. A stable range for quadratic forms over commutative rings. (English) Zbl 0886.11020 J. Pure Appl. Algebra 120, No. 3, 255-280 (1997). The quadratic stable range property is discussed. The ring \(A\) has quadratic stable range \(1\) (\(\text{qsr}(A) = 1\)) if every primitive quadratic form over \(A\) represents a unit. The property is motivated by the ring of holomorphic functions on a connected noncompact Riemann surface. The authors prove the following results:(1) if \(\text{qsr} (A)=1\) then the stable range of \(A\) equals \(1\) and \(\text{Pic} (A)=1\). (2) \(\text{qsr} (A)=1\) iff \(\text{Pic}(T)=1\) for every quadratic \(A\)-algebra \(T\). They also classify quadratic forms over Bezout domains of characteristic not 2 satisfying a very strong approximation property (defined in the paper). This classification applies to the ring of holomorphic functions mentioned above. Reviewer: K.Kozioł (Katowice) Cited in 1 Document MSC: 11E08 Quadratic forms over local rings and fields 11E12 Quadratic forms over global rings and fields 15A63 Quadratic and bilinear forms, inner products 14C22 Picard groups 46E25 Rings and algebras of continuous, differentiable or analytic functions Keywords:quadratic stable range; Picard group; quadratic form; holomorphic function; Riemann surface; Bezout domain PDFBibTeX XMLCite \textit{D. R. Estes} and \textit{R. M. Guralnick}, J. Pure Appl. Algebra 120, No. 3, 255--280 (1997; Zbl 0886.11020) Full Text: DOI References: [1] Baeza, R., Quadratic Forms Over Semilocal Rings, (Lecture Notes in Mathematics, Vol. 655 (1978), Springer: Springer New York) · Zbl 0382.10014 [2] Bass, H., Algebraic K-theory, (Math. Lecture Note Series (1968), Benjamin: Benjamin New York) · Zbl 0248.18025 [3] Estes, D.; Guralnick, R., Module equivalences: local to global when primitive polynomials represent units, J. Algebra, 77, 138-157 (1982) · Zbl 0492.13005 [4] Estes, D.; Ohm, J., Stable range in commutative rings, J. Algebra, 7, 343-362 (1967) · Zbl 0156.27303 [5] Forster, O., Lectures on Riemann Surfaces (1981), Springer: Springer New York [6] Guralnick, R., Matrices and representations of rings of analytic functions and other one dimensional rings, (Visiting Scholars Lectures, Mathematics Series, 15 (1986-1987), Texas Tech. University), 15-35 [7] Guralnick, R., Similarity of holomorphic matrices, Linear Algebra Appl., 99, 85-96 (1988) · Zbl 0647.15006 [8] Kirkwood, B.; McDonald, B., The symplectic group over a ring with one in its stable range, Pacific J. Math., 92, 111-125 (1981) · Zbl 0466.20023 [9] Lang, S., Algebraic Number Theory (1970), Springer: Springer New York · Zbl 0211.38404 [10] McDonald, B.; Hershberger, B., The orthogonal group over a full ring, J. Algebra, 51, 536-549 (1978) · Zbl 0377.15009 [11] O’Meara, O. T., Introduction to Quadratic Forms (1963), Springer: Springer New York · Zbl 0107.03301 [12] Serre, J.-P., Local Fields (1979), Springer: Springer New York [13] Van der Kallen, W., The \(K_2\) of rings with many units, Ann. Sci. École Norm. Sup, 4, 473-515 (1977) · Zbl 0393.18012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.