Stable range and almost stable range

Generating solutions of a linear equation and structure of elements of the Zelisko group

“…, ϕ k are called invariant factors of the matrix A. Since invariant factors in (1) are determined uniquely up to associates, the Smith form of A is defined ambiguously.…”

“…This definition was first given by V. Zelisko [16] for the matrix over polynomial ring F [x] in which F is an algebraic closed field of characteristic 0. The definition of the Zelisko group G Φ over the ring R is independent of the choice of the Smith form Φ of A (see (1)). Indeed, let Φ 1 := ΦΥ in which Υ := diag(ε 1 , .…”

“…The examples of rings of stable range 1.5 are Euclidean rings, principal ideal rings, rings of algebraic integers, rings of integer analytic functions, adequate rings [10, p. 20] and [3]. Note that the commutative rings of stable range 1.5 coincide with rings of almost stable range 1 [1,8].…”

“…If for the matrix A we fix Φ in(1), then the matrices P and Q are also defined ambiguously. As it was shown in [10, Property 2.2, p. 63], the set of left transforming matrices of A coincides with the right coset G Φ P of the Zelisko group G Φ in GL n (R).…”

Generating solutions of a linear equation and structure of elements of the Zelisko group II

“…Examples of rings of stable range 1.5 are Euclidean rings, principal ideal rings, factorial rings, rings of algebraic integers, rings of integer analytic functions, and adequate rings (see [3,4] and [10, p. 21]). Note that the commutative rings of stable range 1.5 coincide with rings of almost stable range 1 (see [1,8]). Finally, certain properties of the Zelisko group G Φ are closely related to a factorizability of the general linear group over the ring R of stable range 1.5 (see [13,Theorem 3,p.…”