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"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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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at franko.lviv.ua

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