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

Irwing Kaplansky

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

Oksana Pihura

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

1. Білявська С. І., Забавський Б. В., Зв’язок адекватних кілець з чистими кільцями // Прикладні проблеми механіки І математики – 2012. – № 8. – С. 28 – 32.
2. 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.
3. HelmerO., The elementary divisor for certain rings without chain condition // Bull. Amer. Math. Soc. – 1943. – 49, № 2 – P. 225 – 236.
4. Larsen M., Lewis W., Shores T., Elementary divisor rings and finitely presented modules // Trans. Amer. Math. Soc. – 1974. –№ 7 – P. 231 – 248.

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