scholarly journals Comaximal factorization in a commutative Bezout ring

2020 ◽  
Vol 30 (1) ◽  
pp. 150-160
Author(s):  
B. V. Zabavsky ◽  
O. Romaniv ◽  
B. Kuznitska ◽  
T. Hlova ◽  
◽  
...  
Keyword(s):  
Elementary Divisor ◽  
Unique Factorization ◽  
Bezout Ring

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

scholarly journals Finite homomorphic images of Bezout duo-domains

2014 ◽  
Vol 6 (2) ◽  
pp. 360-366 ◽  
Author(s):  
O.S. Sorokin
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Global Dimension ◽  
Necessary And Sufficient Conditions ◽  
Stable Range ◽  
Bezout Ring ◽  
Injective Modules ◽  
Elementary Divisor Ring ◽  
Necessary And Sufficient ◽  
Free Element

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.

Rings of dyadic range 1

2019 ◽  
Vol 18 (11) ◽  
pp. 1950206
Author(s):  
Bohdan Zabavsky
Keyword(s):  
Elementary Divisor ◽  
Stable Range ◽  
Bezout Ring ◽  
Elementary Divisor Ring

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.

scholarly journals AMALGAMATION OF RINGS DEFINED BY BÉZOUT-LIKE CONDITIONS

2011 ◽  
Vol 10 (06) ◽  
pp. 1343-1350
Author(s):  
MOHAMMED KABBOUR ◽  
NAJIB MAHDOU
Keyword(s):  
Sufficient Conditions ◽  
Elementary Divisor ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Integral Domains ◽  
Bezout Ring ◽  
Elementary Divisor Ring ◽  
Amalgamated Duplication ◽  
Necessary And Sufficient ◽  
Hermite Ring

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.

Some Criteria for Hermite Rings and Elementary Divisor Rings

1974 ◽  
Vol 26 (6) ◽  
pp. 1380-1383 ◽  
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.

Noncommutative elementary divisor rings

1988 ◽  
Vol 39 (4) ◽  
pp. 349-353
Author(s):  
B. V. Zabavskii
Keyword(s):  
Elementary Divisor

Unique factorization results for semigroups of L-functions

2008 ◽  
Vol 341 (3) ◽  
pp. 517-527 ◽  
Author(s):  
Jerzy Kaczorowski ◽  
Giuseppe Molteni ◽  
Alberto Perelli
Keyword(s):  
Unique Factorization

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

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