On Bézout rings (Q2856603)

Recall that a commutative ring with unit \(R\) is called a Bézout ring if every finitely generated ideal of \(R\) is principle. Let \(A\) be a ring and \(E\) be an \(A\)-module. The trivial ring extension (or the Nagata idealization) of \(A\) by \(E\) is denoted by \(A\propto E\). The main result of the paper under review indicates that: Let \(A\subseteq B\) be two integral domains and let \(R:=A\propto B\). Then, \(R:=A\propto B\) is a Bézout ring if and only if \(A\) is a Bézout domain and \(B\) is the quotient field of \(A\).