Hermite ring

Abstract: Let R be a Bezout ring (a commutative ring in which all finitely generated ideals are principal), and let M be a finitely generated R -module. We will study questions of the following sort: (A) If every localization of M can be generated by n elements, can M itself be generated by n elements? (B) If M 0 R m = Rn for some m, n, is Af necessarily free? (C) If every localization of M has an element with zero annihilator, does M itself have such an element? We will answer these and related questions for various familiar classes of Bezout rings. For example, the answer to (B) is "no" for general Bezout rings but "yes" for Hermite rings (defined below). Also, a Hermite ring is an elementary divisor ring if and only if (A) has an affirmative answer for every module M.

Abstract: Classes of irrational function classes, denoted by AS, that lie between the extreme cases of the disk algebra A and the Hardy space H 1 (D), are considered. The corona theorem holds for AS, and the following properties are shown: AS is an integral domain, but not a B ezout domain, AS is a Hermite ring with stable rank 1, and the Banach algebra AS has topological stable rank 2. Consequences to the coprime factorization of transfer functions and stabilizing controller synthesis using a factorization approach are discussed.

Abstract: The notion of Barbilian domain as introduced and studied by W. Leisner [14], [16] and others is refined here to n-Barbilian domain in a free module of rank n. This leads to results that bear on n-dimensional affine ring geometry. The case of infinite rank is also considered. AMS Classification: 51C05. Keywords: Free module, Barbilian domain, Hermite ring, affine ring geometry.