Every zero adequate ring is an exchange ring. (English. Russian original) Zbl 1276.13009
J. Math. Sci., New York 187, No. 2, 153-156 (2012); translation from Fundam. Prikl. Mat. 17, No. 3, 61-66 (2012).
This paper investigates the connection of adequate and zero adequate rings with exchange rings. It gives a characterization of adequate rings and the relationship between commutative Bezout rings and zero adequate rings.
The authors prove the following theorem: Let \(R\) be a commutative Bezout ring and let a be an adequate element of \(R\). Then the following statements hold
The authors prove the following theorem: Let \(R\) be a commutative Bezout ring and let a be an adequate element of \(R\). Then the following statements hold
- 1.
- \(R/aR\) is an exchange ring;
- 2.
- \(R/aR\) is a clean ring;
- 3.
- \(R/aR\) is a ring of idempotent stable range \(1\);
- 4.
- \(R/aR\) is a PM-ring.
Reviewer: Driss Karim (Nador)
MSC:
13A99 | General commutative ring theory |
13A15 | Ideals and multiplicative ideal theory in commutative rings |
13B02 | Extension theory of commutative rings |
13E05 | Commutative Noetherian rings and modules |
Keywords:
adequate element; adequate ring; Bézout ring; clean ring; exchange ring; PM-ring; zero adequate ringReferences:
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