On bézout domains, elementary divisor rings, and pole assignability

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.

Download Full-text

Gelfand local Bezout domains are elementary divisor rings

Carpathian Mathematical Publications ◽

10.15330/cmp.7.2.188-190 ◽

2015 ◽

Vol 7 (2) ◽

pp. 188-190 ◽

Cited By ~ 1

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.

Download Full-text

Elementary Divisor domains and Bézout domains

Journal of Algebra ◽

10.1016/j.jalgebra.2012.08.020 ◽

2012 ◽

Vol 371 ◽

pp. 609-619 ◽

Cited By ~ 3

Author(s):

Dino Lorenzini

Keyword(s):

Elementary Divisor ◽

Bezout Domains

Download Full-text

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

Carpathian Mathematical Publications ◽

10.15330/cmp.10.2.402-407 ◽

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.

Download Full-text

Some Criteria for Hermite Rings and Elementary Divisor Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-131-8 ◽

1974 ◽

Vol 26 (6) ◽

pp. 1380-1383 ◽

Cited By ~ 3

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.

Download Full-text

Noncommutative elementary divisor rings

Ukrainian Mathematical Journal ◽

10.1007/bf01060766 ◽

1988 ◽

Vol 39 (4) ◽

pp. 349-353

Author(s):

B. V. Zabavskii

Keyword(s):

Elementary Divisor

Download Full-text

Some Theorems On Matrices With Real Quaternion Elements

Canadian Journal of Mathematics ◽

10.4153/cjm-1955-024-x ◽

1955 ◽

Vol 7 ◽

pp. 191-201 ◽

Cited By ~ 63

Author(s):

N. A. Wiegmann

Keyword(s):

Similarity Transformation ◽

Elementary Divisor ◽

Quaternion Matrix ◽

Unitary Similarity ◽

Triangular Form ◽

Quaternion Matrices ◽

Divisor Theory ◽

Real Quaternion ◽

Unitary Similarity Transformation

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

Download Full-text