Bezout Domain Research Articles

The structure of the Smith normal form of the greatest common divisor and the least common multiple of the third order matrices over Bezout domains of stable range 1.5

The structure of the Smith normal form of the greatest common divisor and the least common multiple of the third order matrices over Bezout domains of stable range 1.5

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.

Domains whose ideals meet a universal restriction

Let [Formula: see text] represent a set of proper nonzero ideals [Formula: see text] (respectively, t-ideals [Formula: see text]) of an integral domain [Formula: see text] and let P be a valid property of ideals of D. We say S(D) meets P (denoted [Formula: see text] if each [Formula: see text] is contained in an ideal satisfying P. If S(D) [Formula: see text] [Formula: see text] cannot be controlled. When [Formula: see text] [Formula: see text] [Formula: see text] does not imply I(R) [Formula: see text] while [Formula: see text] [Formula: see text] implies [Formula: see text] [Formula: see text] usually. We say S(D) meets P with a twist [Formula: see text]written [Formula: see text] if each [Formula: see text] is such that, for some [Formula: see text] [Formula: see text] is contained in an ideal satisfying P and study [Formula: see text] as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.

Ideal structure of rings of analytic functions with non-Archimedean metrics

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

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.

Reduction of matrices over simple Ore domains

We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout domains.

Kronecker function rings and power series rings

Let D be an integral domain with quotient field K , X be an indeterminate over D , D [ [ X ] ] be the power series ring over D , and c ( f ) be the ideal of D generated by the coefficients of f ∈ D [ [ X ] ] . We will say that a star operation ∗ on D is a c-star operation if (i) c ( f g ) ∗ = ( c ( f ) c ( g ) ) ∗ for all 0 ≠ f , g ∈ D [ [ X ] ] and (ii) ( A B ) ∗ ⊆ ( A C ) ∗ implies B ∗ ⊆ C ∗ for all nonzero fractional ideals A , B , C of D . Assume that D admits a c-star operation ∗ , and let Kr ( ( D , ∗ ) ) = { f ∕ g ∣ f , g ∈ D [ [ X ] ] , g ≠ 0 , and c ( f ) ⊆ c ( g ) ∗ } . Among other things, we show that Kr ( ( D , ∗ ) ) is a Bezout domain, D is completely integrally closed, the v -operation on D is a c-star operation, and Kr ( ( D , v ) ) is a completely integrally closed Bezout domain. We also show that if V is a rank-one valuation domain, then the v -operation on V is a c-star operation, Kr ( ( V , v ) ) is a rank-one valuation domain, and Kr ( ( V , v ) ) is a DVR if and only if V is a DVR. Using this result, we show that if D is a generalized Krull domain, then Kr ( ( D , v ) ) is a one-dimensional generalized Krull domain.

On Maximal Unipotent Subgroups of a Special Linear Group Over Commutative Ring

It is proved that all maximal unipotent subgroups of a special linear group over commutative ring with identity (such that the factor ring of its modulo primitive radical is a finite direct sum of Bezout domains) are pairwise conjugated and describe one maximal unipotent subgroup of the general linear group (and of a special linear group) over an arbitrary commutative ring with identity.

J-Noetherian Bezout domain which is not of stable range 1

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).

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.

When R(X) and R⟨X⟩ are ω-Euclidean domains

In this paper, it is proved that a domain R is ω-Euclidean if and only if R ⟨ X ⟩ is ω-Euclidean domain. Also, we show that R is a Bezout domain if and only if R is a ring of elementary reduction matrices. Cite as: A. V. Sagan, "When R ( X ) and R ⟨ X ⟩ are ω-Euclidean domains," Prykl. Probl. Mekh. Mat. , Issue 16, 36–38 (2018) (in Ukrainian), https://doi.org/10.15407/apmm2018.16.36-38

Locally adequate duo rings as a generalization case of right adequate rings

A locally right adequate duo-rings have been investigated, the relation between right adequate elements and neat elements in the duo Bezout domain has been shown. It has been proved that the Bezout duo domain is locally right adequate if and only if for any a,b∈R such that aR+bR=R , one of these elements is right adequate. Cite as: A. A. Dmytruk, "Locally adequate duo rings as a generalization case of right adequate rings," Prykl. Probl. Mekh. Mat. , Issue 16, 39–42 (2018) (in Ukrainian), https://doi.org/10.15407/apmm2018.16.39-42

Characterizations of Graded Prüfer $$\star $$ ⋆ -Multiplication Domains, II

Let $$R=\bigoplus _{\alpha \in \Gamma }R_{\alpha }$$ be a graded integral domain and $$\star $$ be a semistar operation on R. For $$a\in R$$ , denote by C(a) the ideal of R generated by homogeneous components of a and for $$f=f_0+f_1X+\cdots +f_nX^n\in R[X]$$ , let $$\mathcal {A}_f:=\sum _{i=0}^nC(f_i)$$ . Let $$N(\star ):=\{f\in R[X]\mid f\ne 0\text { and }\mathcal {A}_f^{\star }=R^{\star }\}$$ . In this paper, we study relationships between ideal theoretic properties of $${\text {NA}}(R,\star ):=R[X]_{N(\star )}$$ and the homogeneous ideal theoretic properties of R. For example, we show that R is a graded Prufer- $$\star $$ -multiplication domain if and only if $${\text {NA}}(D,\star )$$ is a Prufer domain and if and only if $${\text {NA}}(R,\star )$$ is a Bezout domain. We also determine when $${\text {NA}}(R,v)$$ is a PID.

Noether Normalization theorem and dynamical Gröbner bases over Bezout domains of Krull dimension 1

We propose a new version of Noether normalization theorem over Bezout domains of Krull dimension one (the ring Z of integers as main example). We also show that one can avoid branching when computing dynamical Gröbner bases over a Bezout domain of Krull dimension one.

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.

Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring R is a prime ideal of the semigroup $$(R,\cdot )$$ . One of the purposes of this paper is to study, from a topological point of view, the space $${\varvec{\mathcal {S}}}(R)$$ of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, $${\varvec{\mathcal {S}}}(R)$$ is a spectral space (after Hochster), spectral extension of $${{\mathtt {Spec}}}(R)$$ , and that the assignment $$R\mapsto {\varvec{\mathcal {S}}}(R)$$ induces a contravariant functor. We then relate—in the case R is an integral domain—the topology on $${\varvec{\mathcal {S}}}(R)$$ with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between $${\varvec{\mathcal {S}}}(R)$$ and the space $$\varvec{\mathcal {X}}(R)$$ consisting of all nonempty inverse-closed subspaces of $${{\mathtt {Spec}}}(R)$$ , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that $${\varvec{\mathcal {S}}}( R)$$ is a spectral retract of $$\varvec{\mathcal {X}}(R)$$ and we characterize when $${\varvec{\mathcal {S}}}( R)$$ is canonically homeomorphic to $$\varvec{\mathcal {X}}(R)$$ , both in general and when $${{\mathtt {Spec}}}(R)$$ is a Noetherian space. In particular, we obtain that, when R is a Bezout domain, $${\varvec{\mathcal {S}}}( R)$$ is canonically homeomorphic both to $$\varvec{\mathcal {X}}(R)$$ and to the space $$\mathtt {Overr}(R)$$ of the overrings of R (endowed with the Zariski topology). Finally, we compare the space $$\varvec{\mathcal {X}}(R)$$ with the space $${\varvec{\mathcal {S}}}(R(T))$$ of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding $${\varvec{\mathcal {X}}}(R)\hookrightarrow {\varvec{\mathcal {S}}}(R(T))$$ which makes $${\varvec{\mathcal {X}}}(R)$$ a spectral retract of $${\varvec{\mathcal {S}}}(R(T))$$ .

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.

Power series over Noetherian domains, Nagata rings, and Kronecker function rings

Let D be a Noetherian domain, ⁎ be a star operation on D, X be an indeterminate over D, D[[X]] be the power series ring over D, c(f) be the ideal of D generated by the coefficients of f∈D[[X]], and N⁎={f∈D[[X]]|c(f)⁎=D}. Moreover, if ⁎ is e.a.b., then we let Kr((D,⁎))={fg|f,g∈D[[X]],0≠g, and c(f)⊆c(g)⁎}. In this paper, we show that N⁎ is a saturated multiplicative set and Kr((D,⁎)) is a Bezout domain. We then study some ring-theoretic properties of D[[X]]N⁎ and Kr((D,⁎)). For example, we prove that every invertible ideal of D[[X]]N⁎ is principal; dim(Kr((D,b)))=dimv(D); and if V is a valuation overring of D, then Vˆ={fg|f,g∈D[[X]],g≠0, and c(f)V⊆c(g)V} is a valuation overring of D[[X]]; and Kr((D,⁎))=⋂{Vˆ|V is a ⁎-valuation overring of D}.