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

DOI:
https://doi.org/10.1090/S0002-9939-1986-0835863-1

MathSciNet review:
835863

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a principal ideal ring, a symmetric -by- matrix over , a -by- matrix over such that the -by- matrix is primitive. Newman [**2**] 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 [**1**].

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].

**[1]**H. Bass,*-theory and stable algebra*, Inst. Hautes Études Sci. Publ. Math.**22**(1964), 485-544. MR**0174604 (30:4805)****[2]**M. Newman,*Matrix completion theorems*, Proc. Amer. Math. Soc.**94**(1985), 39-45. MR**781052 (86d:15009)****[3]**L. N. Vaserstein,*The stable range of rings and dimension of topological spaces*, Funktsional. Anal. i Prilozhen.**5**(1971), 17-27 (translated in Functional Anal. Appl.). MR**0284476 (44:1701)****[4]**-,*Stabilization for unitary and orthogonal groups over a ring with involution*, Mat. Sb.**81**(1970), 328-352 (translated in Math. USSR-Sb.**10**). MR**0269722 (42:4617)****[5]**-,*Stabilization for the classical groups over rings*, Mat. Sb.**93**(1974), 268-295 (translated in Math. USSR-Sb.**22**). MR**0338208 (49:2974)****[6]**-,*Bass's first stable range conditions*, J. Pure Appl. Algebra**34**(1984), 319-330. MR**772066 (86c:18009)****[7]**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)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0835863-1

Article copyright:
© Copyright 1986
American Mathematical Society