Every zero adequate ring is an exchange ring
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 {\parindent=6mm \begin{itemize}\item[1.] \(R/aR\) is an exchange ring; \item[2.] \(R/aR\) is a clean ring; \item[3.] \(R/aR\) is a ring of idempotent stable range \(1\); \item[4.] \(R/aR\) is a PM-ring. \end{itemize}} The authors conclude the paper by giving the answer of the following question: If \(R\) is a commutative Bezout ring which \(0\) is an adequate element, is \(R\) a zero adequate ring? Theorem. Let \(R\) be a commutative Bezout ring in which \(0\) is an adequate element and \(R\) have finite minimal primes. Then \(R\) is a zero adequate ring.
scientific article; zbMATH DE number 6157002