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.

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.

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.

scholarly journals On strongly pi-regular rings of stable range one

1995 ◽  
Vol 51 (3) ◽  
pp. 433-437 ◽  
Author(s):  
Hua-Ping Yu ◽  
Victor P. Camilo
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Associative Ring ◽  
Stable Range ◽  
Von Neumann ◽  
Stable Range One ◽  
Von Neumann Regular ◽  
Open Question ◽  
Regular Rings

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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