# Commutative Bezout rings in which 0 is adequate is a semi regular

- Written by Oxana Pihura
- Be the first to comment!

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

** **

Abstract:

*A ring R is a commutative ring with nonzero identity.*

*All necessary definitions and facts can be found in [1-4].*

*An element a∈R is called adequate if for any element b∈R we can find such two elements elements r, s∈R that the decomposition a=rs satisfies the following properties:**1) rR+bR=R**2) s'R+bR≠R, for any noninvertible divisor s' of element s.*

**Theorem 1**. A ring R is a commutative ring in which 0 is adequate. Then for **0**∈R/aR an element **0** is adequate.

**Theorem 2**. A ring R is a commutative Bezout ring in which 0 is adequate. Then for any nonzero and any noninvertible alement b∈R there exist an idempotent e∈R, such that be∈J(R) and eR+bR=R.

**Theorem 3**. Semi prime commutative Bezout ring R is a ring in which zero is adequate if and only if R is a regular ring.

**Theorem 4**. A ring R is a commutative Bezout ring. Then the following statements are equivalent:* 1) R is a ring in which zero is adequate;** 2) R is a semi regular ring.*

* *

**References**

- Білявська С. І., Забавський Б. В., Зв’язок адекватних кілець з чистими кільцями // Прикладні проблеми механіки І математики – 2012. – № 8. – С. 28 – 32.
- Забавский Б. В., Билявская С. И., Адекватное в нули кольцо является кольцом со свойством замены // Фундаментальнаяи прикладная математика, 2011/2012. т. – 17, № 3, С. – 61 – 66.
*Helmer**O**.,*The elementary divisor for certain rings without chain condition // Bull. Amer. Math. Soc. – 1943. – 49, № 2 – P. 225 – 236.*Larsen M., Lewis W., Shores T.,*Elementary divisor rings and finitely presented modules // Trans. Amer. Math. Soc. – 1974. –№ 7 – P. 231 – 248.

## Коментарі (0)