Noncommutative elementary divisor rings

1988 ◽  
Vol 39 (4) ◽  
pp. 349-353
Author(s):  
B. V. Zabavskii
Keyword(s):  
1974 ◽  
Vol 26 (6) ◽  
pp. 1380-1383 ◽  
Author(s):  
Thomas S. Shores ◽  
Roger Wiegand

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.


1955 ◽  
Vol 7 ◽  
pp. 191-201 ◽  
Author(s):  
N. A. Wiegmann

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.


2019 ◽  
Vol 18 (08) ◽  
pp. 1950141
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani Abdolyousefi

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.


2018 ◽  
Vol 34 ◽  
pp. 1-17 ◽  
Author(s):  
Lazaros Moysis ◽  
Nicholas Karampetakis

For a given system of algebraic and difference equations, written as an Auto-Regressive (AR) representation $A(\sigma)\beta(k)=0$, where $\sigma $ denotes the shift forward operator and $A\left( \sigma \right) $ a regular polynomial matrix, the forward-backward behavior of this system can be constructed by using the finite and infinite elementary divisor structure of $A\left( \sigma \right) $. This work studies the inverse problem: Given a specific forward-backward behavior, find a family of regular or non-regular polynomial matrices $A\left( \sigma \right) $, such that the constructed system $A\left( \sigma \right) \beta \left( k\right) =0$ has exactly the prescribed behavior. It is proved that this problem can be reduced either to a linear system of equations problem or to an interpolation problem and an algorithm is proposed for constructing a system satisfying a given forward and/or backward behavior.


1973 ◽  
Vol 24 (1) ◽  
pp. 133-141 ◽  
Author(s):  
Melvin Henriksen

1984 ◽  
Vol 12 (24) ◽  
pp. 2987-3003 ◽  
Author(s):  
J.W. Brewer ◽  
C. Naudé ◽  
G. Naudé

1973 ◽  
Vol 16 (3) ◽  
pp. 262-271
Author(s):  
James M. McPherson

The nullity of the Alexander module of the fundamental group of the complement of a knot in S3 was one of the invariants of wild knot type defined and investigated by E. J. Brody in [1], in which he developed a generalised elementary divisor theory applicable to infinitely generated modules over a unique factorisation domain. Brody asked whether the nullity of a knot with one wild point was bounded above by its enclosure genus; for knots in S3, the present author showed in [6] that this was indeed the case. In [7], it was (prematurely) stated by the author that this was also the case for knots k embedded in a 3-manifold M so that H,(M — k) was torsion-free.


1969 ◽  
Vol 76 (4) ◽  
pp. 395 ◽  
Author(s):  
D. B. Lahiri

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