An answer to a question of M. Newman on matrix completion
Author: L. N. Vaserstein
Journal: Proc. Amer. Math. Soc. 97 (1986), 189-196
MSC: Primary 18F25; Secondary 13D15, 15A33, 19B10
MathSciNet review: 835863
Abstract: Let be a principal ideal ring, a symmetric -by- matrix over , a -by- matrix over such that the -by- matrix is primitive. Newman  proved that may be completed (as the first rows) to a symmetric -by- matrix of determinant 1, provided that . He showed that the result is false, in general, if , and he asked to determine all values of such that and the result holds. It is shown here that these values are exactly satisfying .
Moreover, the result is proved for a larger (than the principal ideal rings) class of commutative rings, namely, for the rings satisfying the second stable range condition of Bass .
Also, it is observed that Theorems 2 and 3 of [2, p. 40] proved there for principal ideal rings are true for this larger class of rings, as well as the basic result of [2, p. 39].
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- -, Stabilization for the classical groups over rings, Mat. Sb. 93 (1974), 268-295 (translated in Math. USSR-Sb. 22). MR 0338208 (49:2974)
- -, Bass's first stable range conditions, J. Pure Appl. Algebra 34 (1984), 319-330. MR 772066 (86c:18009)
- L. N. Vaserstein and A. A. Suslin, Serre's problem on projective modules over polynomial rings and algebraic -theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 993-1054 (translated in Math. USSR-Izv. 10). MR 0447245 (56:5560)