A simple extension of a von Neumann regular ring (Q1082379)

Let \(R\) be a commutative von Neumann regular ring and \(R[\alpha]\) be a reduced simple extension of \(R\), i.e. \(R[\alpha]\cong R[x]/I\) with a semi- prime ideal \(I \leq R[x]\) such that \(R\cap I=\{0\}\). The author gives conditions on \(I\) for \(R[\alpha]\) to be quasi-regular. E.g. theorem 6: If C(I) is principal, then \(R(\alpha)\) is quasi-regular. Hence corollary 7: If \(I\) is finitely generated, then \(R[\alpha]\) is quasi- regular. Here a ring is called quasi-regular, if its total ring of quotients is von Neumann regular. If \(f=a_ 0+a_ 1x+a_ 2x^ 2+\dots+a_ nx^ n\in R[x]\), then \(C(f)=(a_ 0,\dots,a_ n)\) is the ideal of \(R\) generated by the \(a_ i\), \(0\leq i\leq n\), and \(C(I)\) is the ideal of \(R\) generated by \(\{C(f): f\in I\}.\) If \(C(I)\) is not principal, the author gives an example to show in this situation \(R[\(\alpha\)]\) need not be quasi-regular.