Commutative Bezout Domain Research Articles

THE MATRIX LINEAR UNILATERAL AND BILATERAL EQUATIONS

In this article the method of solving matrix linear equations over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is found. The criterions of uniqueness of particular solutions of matrix linear equations are determined. The formulas of general solutions of matrix linear equations AX + BY = C and AX + YB = C are deduced.

A Bezout ring with nonzero principal Jacobson radical

In this paper, we study a commutative Bezout domain with nonzero Jacobson radical being a principal ideal. It has been proved that such a Bezout domain is a ring of the stable range 1. As a result, we have obtained that such a Bezout domain is a ring over which any matrix can be reduced to a canonical diagonal form by means of elementary transformations of its rows and columns.

Clear rings and clear elements

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.

Almost zip Bezout domain

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.

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

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.

Bezout Rings of Stable Range 1.5 and the Decomposition of a Complete Linear Group into the Product of its Subgroups

A ring R is called a ring of stable range 1.5 if, for any triple a, b, c ∈ R, c ≠ 0, such that aR + bR + cR = R, there exists r ∈ R such that (a + br)R + cR = R. It is proved that a commutative Bezout domain has a stable range 1.5 if and only if every invertible matrix A can be represented in the form A = HLU, where L, U are elements of the groups of lower and upper unitriangular matrices (triangular matrices with 1 on the diagonal) and the matrix H belongs to the group $$ {\mathrm{G}}_{\Phi}=\left\{H\in {\mathrm{G}\mathrm{L}}_n\left.(R)\right|\exists {H}_1\in {\mathrm{G}\mathrm{L}}_n(R):H\Phi =\Phi {H}_1\right\}, $$ where Φ = diag(φ 1, φ 2, … , φ n ) , φ 1|φ 2| … |φ n , φ n ≠ 0.

Conditions for stable range of an elementary divisor rings

ABSTRACTUsing the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different classes of commutative Bezout domains, in particular, PM*, local Gelfand domains and so on.

Bezout Rings of Stable Range 1.5

A ring R has a stable range 1.5 if, for every triple of left relatively prime nonzero elements a, b, and c in R, there exists r such that the elements a+br and c are left relatively prime. Let R be a commutative Bezout domain. We prove that the matrix ring M 2 (R) has the stable range 1.5 if and only if the ring R has the same stable range.

Geometry of Alternate Matrices over 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.

Commutative domains of elementary divisors and some properties of their elements

We study commutative domains of elementary divisors from the viewpoint of investigation of the structure of invertible matrices that reduce a given matrix to the diagonal form. Some properties of elements of these domains are indicated. We establish conditions, close to the stable-range conditions, under which a commutative Bezout domain is a domain of elementary divisors.

The Matrix Linear Unilateral and Bilateral Equations with Two Variables over Commutative Rings

The method of solving matrix linear equations and over commutative Bezout domains by means of standard form of a pair of matrices with respect to generalized equivalence is proposed. The formulas of general solutions of such equations are deduced. The criterions of uniqueness of particular solutions of such matrix equations are established.

A generalization of the Jaffard–Ohm–Kaplansky Theorem

The well-known Jaffard–Ohm–Kaplansky Theorem states that every abelian l-group can be realized as the group of divisibility of a commutative Bezout domain. To date there is no realization (except in certain circumstances) of an arbitrary, not necessarily abelian, l-group as the group of divisibility of an integral domain. We show that using filters on lattices we can construct a nice quantal frame whose “group of divisibility” is the given l-group. We then show that our construction when applied to an abelian l-group gives rise to the lattice of ideals of any Prufer domain assured by the Jaffard–Ohm–Kaplansky Theorem. Thus, we are assured of the appropriate generalization of the Jaffard–Ohm–Kaplansky Theorem.

Universal localizations embedded in power-series rings

Let R be a ring, let F be a free group, and let X be a basis of F. Let †: RF ! R denote the usual augmentation map for the group ring RF, let X@ := fx i 1 j x 2 Xg µ RF, let § denote the set of matrices over RF that are sent to invertible matrices by †, and let (RF)§ i1 denote the universal localization of RF at §. A classic result of Magnus and Fox gives an embedding of RF in the power-series ring RhhX@ii. We show that if R is a commutative Bezout domain, then the division closure of the image of RF in RhhX@ii is a universal localization of RF at §. We also show that if R is a von Neumann regular ring or a commutative Bezout domain, then (RF)§ i1 is stably flat as an RF-ring, in the sense of

Generalized adequate rings

We introduce a new class of rings of elementary divisors which generalize adequate rings. We show that the problem of whether every commutative Bezout domain is a domain of elementary divisors reduces to the case where the domain contains only trivial adequate elements (namely, the identities of the domain).