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 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 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.

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

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.

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.

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.

scholarly journals Gelfand local Bezout domains are elementary divisor rings

2015 ◽  
Vol 7 (2) ◽  
pp. 188-190 ◽  
Author(s):  
B.V. Zabavsky ◽  
O.V. Pihura
Keyword(s):  
Elementary Divisor ◽  
Local Rings ◽  
Bezout Domains

We introduce the Gelfand local rings. In the case of commutative Gelfand local Bezout domains we show that they are an elementary divisor domains.

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].

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