elementary divisor
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2021 ◽  
Vol 55 (1) ◽  
pp. 3-9
Author(s):  
B. V. Zabavsky ◽  
O. V. Domsha ◽  
O. M. Romaniv

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.


Author(s):  
Sahar Jaafar Mahmood ◽  
Nesir Rasool Mahmood ◽  
Dhirgam Allawy Hussein

In this article, we find the cyclic decomposition of the finite abelian factor group AC(G)=\bar{R}(G)/T(G), where G=Q_{2m} and m is an even number and Q_{2m} is the quaternion group of order 4m. (The group of all Z-valued generalized characters of G over the group of induced unit characters from all cyclic subgroups of G). We find that the cyclic decomposition AC(Q_{2m}) depends on the elementary divisor of m. We have found that if m= p_{1}^{r_1} \cdot p_{2}^{r_2} \cdots p_{n}^{r_n} \cdot 2^h, p_i are distinct primes, then: AC(Q_{2m})=\bigoplus_{i=1}^{(r_1+1)(r_2+1)\cdots(r_n+n)(h+2)-1}C_2. Moreover, we have also found the general form of Artin characters table Ar(Q_{2m}) when m is an even number.


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 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 10 (2) ◽  
pp. 402-407
Author(s):  
B.V. Zabavsky ◽  
O.M. Romaniv

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.


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