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

Irwing Kaplansky

# Combining local and avoidable rings

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

Olga Domsha

The Lviv Regional Institute of Public Administration

Abstract:

All rings R are considered to be commutative and have a nontrivial identity.

Definition. An element aR is said to be an avoidable element in R if for every elements b, c such that aR+bR+cR=R we have a=rs where rR+bR=R, sR+cR=R, rR+sR=R. A ring R is said to be a local avoidable if for every element aR at least one of elements a or 1-a is an avoidable element. A ring R is said to be of avoidable range 1 if the condition aR+bR=R implies that there exists an element tR such that a+bt is an avoidable element.

Theorem 1. Any VNL ring is a local avoidable ring. Any adequate ring is a local avoidable ring. Any local adequate ring is a local avoidable ring.

Theorem 2. Any local avoidable ring is a ring of avoidable range 1.

Theorem 3. A commutative Bezout domain of avoidable range 1 is an elementary division ring

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