On the stable range of matrix rings

Abstract

[] is the integer part of the number m. Therefore, if the stable range of the ring R is equal to 1 or 2, then the stable range of the matrix ring Mn R ( , ) is also equal to 1 or 2, respectively. Among the rings of finite stable range, we select a class of rings of elementary divisors introduced by Kaplansky in [6]. Most of the known classes of rings of elementary divisors strongly depend on the ascending chain conditions for ideals. The first example of the ring of elementary divisors without ascending chain conditions for ideals (a ring of analytic functions) was proposed by Wedderburn [7]. This example enabled Helmer to introduce a new class of rings of elementary divisors called adequate rings [8]. It is known [9] that the stable range of any adequate ring does not exceed 2. In many cases, it is equal to 1. In the present paper, we establish the conditions under which the stable range of an adequate ring is equal to 1. By using the standard form of a pair of matrices relative to the generalized equivalence in the matrix ring Mn R ( , ) of order n over the adequate ring R, we select a class of matrices of stable range 1 in the case where the stable range of the ring R can be greater than 1.