Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
The Lviv Regional Institute of Public Administration
National Academy of Public Administration of President of Ukraine
All rings R are considered to be commutative and have a nontrivial identity.
Definition. An element a∈R 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 a∈R 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 t∈R 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