A Sharp Bézout Domain is an Elementary Divisor Ring

2014 ◽  
Vol 66 (2) ◽  
pp. 317-321 ◽  
Author(s):  
B. V. Zabavs’kyi
Keyword(s):  
Elementary Divisor ◽  
Bezout Domain ◽  
Elementary Divisor Ring

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.

scholarly journals Clear rings and clear elements

2021 ◽  
Vol 55 (1) ◽  
pp. 3-9
Author(s):  
B. V. Zabavsky ◽  
O. V. Domsha ◽  
O. M. Romaniv
Keyword(s):  
Regular Element ◽  
Associative Ring ◽  
Elementary Divisor ◽  
Bezout Domain ◽  
Elementary Divisor Ring

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.

Some Criteria for Hermite Rings and Elementary Divisor Rings

1974 ◽  
Vol 26 (6) ◽  
pp. 1380-1383 ◽  
Author(s):  
Thomas S. Shores ◽  
Roger Wiegand
Keyword(s):  
Triangular Matrix ◽  
Elementary Divisor ◽  
Diagonal Matrix ◽  
Finitely Generated ◽  
Elementary Divisor Ring ◽  
Hermite Ring

Recall that a ring R (all rings considered are commutative with unit) is an elementary divisor ring (respectively, a Hermite ring) provided every matrix over R is equivalent to a diagonal matrix (respectively, a triangular matrix). Thus, every elementary divisor ring is Hermite, and it is easily seen that a Hermite ring is Bezout, that is, finitely generated ideals are principal. Examples have been given [4] to show that neither implication is reversible.

scholarly journals Finite homomorphic images of Bezout duo-domains

2014 ◽  
Vol 6 (2) ◽  
pp. 360-366 ◽  
Author(s):  
O.S. Sorokin
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Global Dimension ◽  
Necessary And Sufficient Conditions ◽  
Stable Range ◽  
Bezout Ring ◽  
Injective Modules ◽  
Elementary Divisor Ring ◽  
Necessary And Sufficient ◽  
Free Element

It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

A Stable Range of Class Full Matrices over Elementary Divisor Ring

2014 ◽  
Vol 66 (5) ◽  
pp. 792-795
Author(s):  
B. V. Zabavs’kyi ◽  
B. M. Kuznits’ka
Keyword(s):  
Elementary Divisor ◽  
Stable Range ◽  
Elementary Divisor Ring

Rings of dyadic range 1

2019 ◽  
Vol 18 (11) ◽  
pp. 1950206
Author(s):  
Bohdan Zabavsky
Keyword(s):  
Elementary Divisor ◽  
Stable Range ◽  
Bezout Ring ◽  
Elementary Divisor Ring

In this paper, we introduced the concept of a ring of a right (left) dyadic range 1. We proved that a Bezout ring of right (left) dyadic range 1 is a ring of stable range 2. And we proved that a commutative Bezout ring is an elementary divisor ring if and only if it is a ring of dyadic range 1.

scholarly journals AMALGAMATION OF RINGS DEFINED BY BÉZOUT-LIKE CONDITIONS

2011 ◽  
Vol 10 (06) ◽  
pp. 1343-1350
Author(s):  
MOHAMMED KABBOUR ◽  
NAJIB MAHDOU
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Integral Domains ◽  
Bezout Ring ◽  
Elementary Divisor Ring ◽  
Amalgamated Duplication ◽  
Necessary And Sufficient ◽  
Hermite Ring

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of notions elementary divisor ring, Hermite ring and Bézout ring to the amalgamation A ⋈f J. We provide necessary and sufficient conditions for A ⋈f J to be an elementary divisor ring where A and B are integral domains. In this case it is shown that A ⋈f J is an Hermite ring if and only if it is a Bézout ring. In particular, we study the transfer of the previous notions to the amalgamated duplication of a ring A along an A-submodule E of Q(A) such that E2 ⊆ E.

scholarly journals Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals

2018 ◽  
Vol 10 (2) ◽  
pp. 402-407
Author(s):  
B.V. Zabavsky ◽  
O.M. Romaniv
Keyword(s):  
Prime Ideal ◽  
Nonzero Element ◽  
Elementary Divisor ◽  
Maximal Ideals ◽  
The Matrix ◽  
Elementary Divisor Ring ◽  
Finite Set ◽  
Invertible Matrices ◽  
Bezout Domains ◽  
Diagonal Reduction

We investigate   commutative Bezout domains in which any nonzero prime  ideal is contained in a finite set of maximal ideals. In particular, we have described the class of such rings, which are  elementary divisor rings. A ring $R$ is called an elementary divisor ring if every matrix over $R$ has a canonical diagonal reduction (we say that a matrix $A$ over $R$ has a canonical diagonal reduction  if for the matrix $A$ there exist invertible matrices $P$ and $Q$ of appropriate sizes and a diagonal matrix $D=\mathrm{diag}(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r,0,\dots,0)$ such that  $PAQ=D$  and $R\varepsilon_i\subseteq R\varepsilon_{i+1}$ for every $1\le i\le r-1$). We proved that a commutative Bezout domain $R$ in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element $a\in R$  the ideal $aR$ a decomposed into a product $aR = Q_1\ldots Q_n$, where  $Q_i$ ($i=1,\ldots, n$) are pairwise comaximal ideals and $\mathrm{rad}\,Q_i\in\mathrm{spec}\, R$,  is an elementary divisor ring.

Noncommutative elementary divisor rings

1988 ◽  
Vol 39 (4) ◽  
pp. 349-353
Author(s):  
B. V. Zabavskii
Keyword(s):  
Elementary Divisor

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.

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