# Combining local and avoidable rings

- Written by B. Kuznitska, O. Domsha
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*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*

**Abstract:**

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*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*

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