Related papers: Rings of the right (left) almost stable range 1
In this paper we introduced the concept of a ring of stable range 2 which has square stable range 1. We proved that a Hermitian ring $R$ which has (right) square stable range 1 is an elementary divisor ring if and only if $R$ is a duo ring…
We present some variants of the Kaplansky condition for a K-Hermite ring $R$ to be an elementary divisor ring; for example, a commutative K-Hermite ring $R$ is an EDR iff for any elements $x,y,z\in R$ such that $(x,y)=(1)$, there exists an…
We constuct the theory of diagonalizability for matrices over Bezout rings of stable range 1 with the Kazimirsky condition. It is shown that a ring of stable range 1 with the right (left) Kazimirsky condition is an elementary divisor ring…
A commutative ring $R$ is J-stable provided that for any $a\not\in J(R)$, $R/aR$ has stable range one. A ring $R$ is called an elementary divisor ring if every $m\times n$ matrix over $R$ admits diagonal reduction. We prove that a J-stabe…
We present some new conditions for a B$\acute{e}$zout ring to be an elementary divisor ring. We prove, in this note, that a B$\acute{e}$zout ring $R$ is feckly zero-adequate if and only if $R/J(R)$ is regular if and only if $R/J(R)$ is…
A commutative ring R is locally stable provided that for any $a,b\in R$ such that $aR+bR=R$, there exist some $y\in R$ such that $R/(a+by)R$ has stable range 1.For a Bezout ring $R$, we prove that $R$ is an elementary divisor ring if and…
A ring $R$ is an elementary divisor ring if every matrix over $R$ admits a diagonal reduction. We further explore various stable like conditions on a bezout duo-domain under which it is an elementary divisor domain. Many known results are…
We introduce noncommutative rings with $DK$-property (Dubrovin-Komarnytsky's property) and investigate elementary divisor rings with such property. Mostly we pay attention to these kinds of noncommutative rings which have stable range $1$.…
A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of…
Using the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different…
We show that a ring R has stable range one if and only if every left unit lifts modulo every left principal ideal. We also show that a left quasi-morphic ring has stable range one if and only if it is left uniquely generated. Thus we answer…
A ring element $\,a\in R\,$ is said to be of {\it right stable range one\/} if, for any $\,t\in R$, $\,aR+tR=R\,$ implies that $\,a+t\,b\,$ is a unit in $\,R\,$ for some $\,b\in R$. Similarly, $\,a\in R\,$ is said to be of {\it left stable…
This article deals mostly with the following question: when is the classical ring of quotients of a commutative ring a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary…
Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and…
We explore elementary matrix reduction over certain rings characterized by their localizations. Let $R$ be a locally stable ring, we prove that $R$ is an elementary divisor ring if and only if $R$ is a Bezout ring. Elementary matrix…
It is shown that a commutative B\'ezout ring $R$ with compact minimal prime spectrum is an elementary divisor ring if and only if so is $R/L$ for each minimal prime ideal $L$. This result is obtained by using the quotient space…
A ring R is said to be of stable range 1.5 if for each a, b from R and nonzero c from R satisfying aR + bR + cR = R there exists r from R such that (a + br)R + cR = R. Let R be a commutative domain in which all finitely generated ideals are…
Using the concept of ring diadic range 1 we proved that a commutative Bezout ring is an elementary divisor ring iff it is a ring diadic range 1.
We proved that in J-Noetherian Bezout domain which is not a ring of stable range 1 exists a nonunit adequate element (element of almost stable range 1
We introduce the concept of rings of simple range 2. Based on this concept, we build a theory diagonal reduction of matrices over Bezout domain. In particular we show that invariant Bezout domain is an elementary divisor ring if and only if…