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Congruence Properties of Zariski‐Dense Subgroups I
This paper deals with the following general situation: we are given an algebraic group G defined over a number field K, and a subgroup F of the group G(K) of K-rational points of G. Then what shouldExpand
Bass's first stable range condition
Let R be an associative ring with unit. Bass’s lowest (the first) stable range condition [l] asserts the following: if a and b in R satisfy Ra + R& = R, then there exists C in R with a + tb leftExpand
Absolute stable rank and Witt cancellation for noncommutative rings
Stable range condi t ions on a ring R were devised by H. Bass in order to determine values of n for which every matr ix in GL,(R) can be row reduced (by addi t ion operat ions with coefficients fromExpand
ON THE GROUP SL2 OVER DEDEKIND RINGS OF ARITHMETIC TYPE
It is proved that the group of matrices of order two with determinant 1 over a Dedekind ring of arithmetic type is generated by elementary matrices if there are infinitely many invertible elements inExpand
ON THE STABILIZATION OF THE GENERAL LINEAR GROUP OVER A RING
We are concerned with the theory of matrix determinants (more precisely, the Whitehead determinant) over any associative ring from the algebraic point of view, i.e. K1-theory. The main goal of theExpand
SERRE'S PROBLEM ON PROJECTIVE MODULES OVER POLYNOMIAL RINGS, AND ALGEBRAIC $ K$-THEORY
We obtain new results on the freeness of projective modules over polynomial rings. In particular we prove the freeness of all projective modules over polynomial rings in five variables withExpand
Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups
A polynomial parametrization for the group of integer two-by-two matrices with determinant one is given, solving an old open problem of Skolem and Beurkers. It follows that, for many DiophantineExpand
Commutators and companion matrices over rings of stable rank 1
Abstract We consider the group GL n A of all invertible n by n matrices over a ring A satisfying the first Bass stable range condition. We prove that every matrix is similar to the product of a lowerExpand
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