Über die Primideale differenzierbarer und integrierbarer Funktionen. (On the prime ideals of differentiable and integrable functions) (Q5966435)

A commutative ring R is called a pc-ring if all prime ideals of R containing a given prime ideal form a chain under inclusion. If R has a unit and every prime ideal contains a unique minimal prime ideal then R is a mp-ring. These are the main results of the paper: (1) All rings \(C^ k({\mathbb{R}}^ n)\) of those real-valued mappings on \({\mathbb{R}}^ n\) which possess continuous partial derivates of order \(\leq k\) satisfy the pc-condition. (2) For all \(n\geq 2\), the ring \(C^{\infty}({\mathbb{R}}^ n):=\cap_{k}C^ k({\mathbb{R}}^ n)\) fails to be a pc-ring (here the case \(n=1\) remains open). (3) Let (X,\({\mathcal B},\mu)\) be a measure space and R(\(\mu)\) the ring of all bounded and (with respect to \(\mu)\) integrable functions \(f:X\to {\mathbb{R}}.\) R(\(\mu)\) is a pc-ring. If \(\mu (X)<\infty\) then R(\(\mu)\) has the mp-property too. (4) The ring R(Q) of all Riemann-integrable functions \(f:Q\to {\mathbb{R}}\) on a closed cube \(Q\subseteq {\mathbb{R}}^ n\) is a pc-ring but not an mp-ring.