Geometry of Alternate Matrices over Bezout Domains

2013 ◽  
Vol 20 (02) ◽  
pp. 197-214 ◽  
Author(s):  
Liping Huang ◽  
Yingchun Li ◽  
Kang Zhao
Keyword(s):  
Bezout Domain ◽  
Bezout Domains

Let R be a commutative Bezout domain. Denote by [Formula: see text] the set of all n × n alternate matrices over R. This paper discusses the adjacency preserving bijective maps in both directions on [Formula: see text], and extends Liu's theorem on the geometry of alternate matrices over a field to the case of a Bezout domain.

Elementary matrix reduction over Bézout domains

2019 ◽  
Vol 18 (08) ◽  
pp. 1950141
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani Abdolyousefi
Keyword(s):  
Elementary Divisor ◽  
Bezout Domain ◽  
Elementary Matrix ◽  
Matrix Reduction ◽  
Elementary Divisor Ring ◽  
Bezout Domains ◽  
Diagonal Reduction

A ring [Formula: see text] is an elementary divisor ring if every matrix over [Formula: see text] admits a diagonal reduction. If [Formula: see text] is an elementary divisor domain, we prove that [Formula: see text] is a Bézout duo-domain if and only if for any [Formula: see text], [Formula: see text] such that [Formula: see text]. We explore certain stable-like conditions on a Bézout domain under which it is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

Overrings of Bezout Domains

1973 ◽  
Vol 16 (4) ◽  
pp. 475-477 ◽  
Author(s):  
Raymond A. Beauregard
Keyword(s):  
Local Ring ◽  
Left Ideal ◽  
Quotient Ring ◽  
Valuation Domain ◽  
Quotient Field ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Chain Conditions ◽  
Bezout Domains

In [2] Brungs shows that every ring T between a principal (right and left) ideal domain R and its quotient field is a quotient ring of R. In this note we obtain similar results without assuming the ascending chain conditions. For a (right and left) Bezout domain R we show that T is a quotient ring of R which is again a Bezout domain; furthermore Tis a valuation domain if and only if T is a local ring.

Bezout Domains and Rings with a Distributive Lattice of Right Ideals

1986 ◽  
Vol 38 (2) ◽  
pp. 286-303 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Distributive Lattice ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Integral Domains ◽  
Bezout Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains

It is the purpose of this paper to discuss a construction of right arithmetical (or right D-domains in [5]) domains, i.e., integral domains R for which the lattice of right ideals is distributive (see also [3]). Whereas the commutative rings in this class are precisely the Prüfer domains, not even right and left principal ideal domains are necessarily arithmetical. Among other things we show that a Bezout domain is right arithmetical if and only if all maximal right ideals are two-sided.Any right ideal of a right noetherian, right arithmetical domain is two-sided. This fact makes it possible to describe the semigroup of right ideals in such a ring in a satisfactory way; [3], [5].

scholarly journals Notes on topological rings

2013 ◽  
Vol 29 (2) ◽  
pp. 267-273
Author(s):  
MIHAIL URSUL ◽  
◽  
MARTIN JURAS ◽  
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Compact Ring ◽  
Ring Topology ◽  
Bezout Domain ◽  
Totally Bounded ◽  
Ideal Domain ◽  
Nilpotent Ring ◽  
Trivial Multiplication

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

Bezout domains and skew polynomials

2000 ◽  
Vol 55 (5) ◽  
pp. 1005-1006 ◽  
Author(s):  
A A Tuganbaev
Keyword(s):  
Skew Polynomials ◽  
Bezout Domains

Kronecker function rings of ring extensions

2018 ◽  
Vol 17 (02) ◽  
pp. 1850021
Author(s):  
Lokendra Paudel ◽  
Simplice Tchamna
Keyword(s):  
General Construction ◽  
Bezout Domain ◽  
Star Operation ◽  
Mild Assumption ◽  
Function Ring ◽  
Ring Extensions

The classical Kronecker function ring construction associates to a domain [Formula: see text] a Bézout domain. Let [Formula: see text] be a subring of a ring [Formula: see text], and let ⋆ be a star operation on the extension [Formula: see text]. In their book [Manis Valuations and Prüfer Extensions II, Lectures Notes in Mathematics, Vol. 2103 (Springer, Cham, 2014)], Knebusch and Kaiser develop a more general construction of the Kronecker function ring of [Formula: see text] with respect to ⋆. We characterize in several ways, under relatively mild assumption on [Formula: see text], the Kronecker function ring as defined by Knebusch and Kaiser. In particular, we focus on the case where [Formula: see text] is a flat epimorphic extension or a Prüfer extension.

scholarly journals Almost Beźout Domains .II.

1994 ◽  
Vol 167 (3) ◽  
pp. 547-556 ◽  
Author(s):  
D.D. Anderson ◽  
K.R. Knopp ◽  
R.L. Lewin
Keyword(s):  
Bezout Domains

On bézout domains, elementary divisor rings, and pole assignability

1984 ◽  
Vol 12 (24) ◽  
pp. 2987-3003 ◽  
Author(s):  
J.W. Brewer ◽  
C. Naudé ◽  
G. Naudé
Keyword(s):  
Elementary Divisor ◽  
Bezout Domains
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