bezout domain
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2021 ◽  
Vol 55 (1) ◽  
pp. 3-9
Author(s):  
B. V. Zabavsky ◽  
O. V. Domsha ◽  
O. M. Romaniv

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear.In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.


2020 ◽  
Vol 53 (2) ◽  
pp. 115-118
Author(s):  
O.M. Romaniv ◽  
B.V. Zabavsky
Keyword(s):  

J. Zelmanowitz introduced the concept of a ring, which we call a zip ring. In this paper we characterize a commutative Bezout domain whose finite homomorphic images are zip rings modulo its nilradical.


2019 ◽  
Vol 19 (10) ◽  
pp. 2050187
Author(s):  
Bohdan Zabavsky ◽  
Oleh Romaniv
Keyword(s):  

We prove that in any [Formula: see text]-Noetherian Bezout domain which is not of stable range 1, there exists a nonunit adequate element (element of almost stable range 1).


2019 ◽  
Vol 18 (08) ◽  
pp. 1950141
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani Abdolyousefi

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.


2018 ◽  
Vol 17 (02) ◽  
pp. 1850021
Author(s):  
Lokendra Paudel ◽  
Simplice Tchamna

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.


2014 ◽  
Vol 79 (01) ◽  
pp. 296-305 ◽  
Author(s):  
GENA PUNINSKI ◽  
CARLO TOFFALORI

Abstract We describe the Ziegler spectrum of a Bézout domain B=D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.


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

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.


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