Coherence and power series rings over pseudo-Bezout domains (Q1087594)

A commutative ring R is said to be coherent when every finitely generated ideal is finitely presented. In this note R is an integral domain in which case coherence is equivalent to finite generation of the intersection of any two finitely generated ideals. The purpose of the note is to produce classes of coherent domains R such that R[[X]] is not coherent, and the starting point seems to be a result of \textit{S. Jøndrup} and \textit{L. W. Small} [Math. Scand. 35, 21-24 (1974; Zbl 0298.13015)] to the effect that if R is a valuation domain of rank \(>1\) then R[[X]] is not coherent. This extended to the case where R is pseudo- Bézout with non-archimedean group of divisibility G(R). The case of G(R) archimedean is also considered, and a second theorem implies that if R is a valuation domain with G(R) archimedean and a proper dense subgroup of \({\mathbb{R}}\) then R[[X]] is not coherent. This leaves as an open question: is R[[X]] coherent when R is a valuation domain and \(G(R)={\mathbb{R}}\) ?