"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

# Bezout rings stable range and its generalizations

Vasylyna Bokhonko

Department of Algebra and Logic

Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv

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 a2+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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