Distributive elementary divisor domains (Q2639140)

An elementary divisor ring or ED-ring is a ring over which every matrix admits a diagonal reduction [cf. \textit{I. Kaplansky}, Trans. Am. Math. Soc. 66, 464-491 (1949; Zbl 0036.01903)]. The authors show that in an ED-ring \(R\) for which every maximal right ideal is two-sided, for every \(a\in R\) there exists \(a^*\) such that \(RaR=a^*R=Ra^*\). If moreover, all zero divisors lie in the Jacobson radical, then \(R\) is a duo-ring. As a consequence they deduce that an ED-domain with distributive right-ideal-lattice is a duo-domain. They also generalize a result proved in the commutative case by \textit{M. Henriksen} [Mich. Math. J. 3, 159-163 (1956; Zbl 0073.02301)] by showing that a distributive Bezout domain \(R\) is an ED-ring if and only if \(R/J(R)\) is an ED-ring.