# Bezout rings stable range and its generalizations

- Written by Vasylyna Bokhonko
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Department of Algebra and Logic* Faculty of Mechanics and Mathematics** Ivan Franko National University of L'viv*

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*Abstract:*

**Theorem 1.*** Let R be a Bezout duo-domain. Then the following conditions are equivalent:**1) R is an elementary divisor duo-domain;**2) For every elements x, y, z∈R such that xR+yR=R there exists an element λ∈R such that x+λy=vu, where uR+zR=R, vR+(1-z)R=R.*

**Definition 1. ***We say that a domain R satisfies the condition Z if whenever RaR=R then a is a finite element.*

**Definition 2. ***We say that a ring R satisfies Dubrovin condition if for any nonzero element a∈R there exists an element α∈R such that RaR=αR=Rα.*

**Theorem 2.** *Let R be a Bezout domain of stable range one satisfying Dubrovin and Z conditions. Then R is an elementary divisor ring.*

**Definition 3.** *A commutative Bezout ring R is a Toeplitz ring if for any elements a, b∈R there exists an invertible Toeplitz matrix T such that (a,b)T=(d,0).*

**Definition 4.** *A ring R is called a ring of unit square stable range one if whenever aR+bR=R then there exists a unit t such that a ^{2}+bt is an invertible element in R.*

**Theorem 3 .**

*Let R be a commutative Hermite ring of unit square stable range one. Then any 2x2 matrix over R can be reduced to the diagonal form multiplying by some invertible Toeplitz matrices.*

**Definition 5.** *Let R be a commutative Bezout domain. The element a∈R\{0} is said to be a semipotent element if for any b∈R there exists a decomposition a=r·s where rR+bR=R and rR+sR=R.*

**Theorem 4.** *Let R be a commutative Bezout domain. Then a is a semipotent element if and only if R/aR is a semipotent ring.*

**Definition 6.*** Let R be a Bezout duo-domain. We say that R is a ring of Gelfand range one if for some elements a,b∈R such that aR+bR=R there exists element r∈R such that a+br is a Gelfand element.*

**Theorem 5.** *Let R be a Bezout duo-domain of Gelfand range one. Then R is an elementary divisor ring.*

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