Bézout rings with almost stable range 1 Warren
نویسنده
چکیده
Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362–365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1) (1974) 231–248], it was also proved that if a Hermite ring satisfies (N), then it is an elementary divisor ring. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that Bézout rings whose proper homomorphic images all have stable range 1 (in particular, neat rings) are elementary divisor rings. c © 2007 Elsevier B.V. All rights reserved. MSC: Primary: 13F99; secondary: 13E15; 06F20
منابع مشابه
Bézout Rings with Almost Stable Range 1 are Elementary Divisor Rings
Abstract. In this article we revisit a problem regarding Bézout domains, namely, whether every Bézout domain is an elementary divisor domain. Elementary divisor domains where defined by Kaplansky [13] and generalized to rings with zero-divisors by Gillman and Henriksen [7]. Later, in [14] it was shown that a domain R is an elementary divisor domain if and only if every finitely presented R-modu...
متن کاملNeat Rings
A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the last 30 years several characterizations of commutative clean rings have been given. We have compiled a thorough list, including some new equivalences, in hopes that in the future there will be a better understanding of this interesting class of rings. One of the fundamental properties of clean rings i...
متن کاملFactorial Rings and Diagonal Reduction of Matrices
The class of Bézout factorial rings is introduced and characterized. Using the factorial properties of such a ring R, and given a n×m matrix A over R, we find P ∈ GL(n, R) and Q ∈ GL(m, R) such that PAQ is diagonal with every element in the diagonal dividing the following one. Key-words: Ring, Bézout, principal, factorization, reduction of matrices.
متن کاملOn the Bézout equation in the ring of periodic distributions
The aim of this short note is to study some algebraic and topological questions associated with the “Bézout equation” b1a1 + · · · + bNaN = e, where bi , ai (1 ≤ i ≤ N) are elements of the commutative unital topological ring (DA(R), +, *, TD′A(Rd)), defined below, and e denotes the identity element (which will be the locally finite sum of Dirac distributions placed at a lattice formed by the pe...
متن کاملElastic Buckling Analysis of Ring and Stringer-stiffened Cylindrical Shells under General Pressure and Axial Compression via the Ritz Method
Elastic stability of ring and stringer-stiffened cylindrical shells under axial, internal and external pressures is studied using Ritz method. The stiffeners are rings, stringers and their different arrangements at the inner and outer surfaces of the shell. Critical buckling loads are obtained using Ritz method. It has been found that the cylindrical shells with outside rings are more stable th...
متن کامل