bezout ring
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2020 ◽  
Vol 30 (1) ◽  
pp. 150-160
Author(s):  
B. V. Zabavsky ◽  
O. Romaniv ◽  
B. Kuznitska ◽  
T. Hlova ◽  
◽  
...  

We study an analogue of unique factorization rings in the case of an elementary divisor domain.


2019 ◽  
Vol 18 (11) ◽  
pp. 1950206
Author(s):  
Bohdan Zabavsky

In this paper, we introduced the concept of a ring of a right (left) dyadic range 1. We proved that a Bezout ring of right (left) dyadic range 1 is a ring of stable range 2. And we proved that a commutative Bezout ring is an elementary divisor ring if and only if it is a ring of dyadic range 1.


2019 ◽  
Vol 47 (12) ◽  
pp. 5392-5397
Author(s):  
Bohdan Zabavsky ◽  
Oleh Romaniv
Keyword(s):  

2015 ◽  
Vol 58 (3) ◽  
pp. 617-635
Author(s):  
O. S. SOROKIN

AbstractThe K-theoretical aspect of the commutative Bezout rings is established using the arithmetical properties of the Bezout rings in order to obtain a ring of all Smith normal forms of matrices over the Bezout ring. The internal structure and basic properties of such rings are discussed as well as their presentations by the Witt vectors. In a case of a commutative von Neumann regular rings the famous Grothendieck group K0(R) obtains the alternative description.


2014 ◽  
Vol 6 (2) ◽  
pp. 360-366 ◽  
Author(s):  
O.S. Sorokin

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.


2011 ◽  
Vol 10 (06) ◽  
pp. 1343-1350
Author(s):  
MOHAMMED KABBOUR ◽  
NAJIB MAHDOU

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of notions elementary divisor ring, Hermite ring and Bézout ring to the amalgamation A ⋈f J. We provide necessary and sufficient conditions for A ⋈f J to be an elementary divisor ring where A and B are integral domains. In this case it is shown that A ⋈f J is an Hermite ring if and only if it is a Bézout ring. In particular, we study the transfer of the previous notions to the amalgamated duplication of a ring A along an A-submodule E of Q(A) such that E2 ⊆ E.


2010 ◽  
Vol 62 (1) ◽  
pp. 151-154
Author(s):  
A. I. Hatalevych
Keyword(s):  

2007 ◽  
Vol 06 (05) ◽  
pp. 789-799 ◽  
Author(s):  
V. CAMILLO ◽  
W. K. NICHOLSON

A ring R is called left morphic if R/Ra ≅ l (a) for each a ∈ R, equivalently if there exists b ∈ R such that Ra = l (b) and l (a) = Rb. In this paper, we ask only that b and c exist such that Ra = l (b) and l (a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bézout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring.


2005 ◽  
Vol 71 (1) ◽  
pp. 113-119
Author(s):  
K. Samei

A ring is called Bezout if every finitely generated ideal is principal. We show that every ideal of a commutative Bezout ring R is flat if and only if every submodule of a free R-module is flat. Using this theorem we obtain Neville's theorem.


1991 ◽  
Vol 22 (4) ◽  
pp. 371-375
Author(s):  
BORIS LAVRIČ
Keyword(s):  

Let $A$ be an Archimedean uniformly complete unital $f$-algebra.It is proved that the following conditions are equivalent: (1) $A$ is a Bezout ring; (2) $A$ is a PF-ring; (3) Every ideal of $A$ is flat; (4) Every submodule of a free $A$-module is flat. This extends a result by C. Neville on algebras of type $C(X)$.


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