Direct limits of Cohen-Macaulay rings (Q2443283)

In this interesting paper, the authors study the Cohen- Macaulay properties of a direct limit of a direct system of Cohen-Macaulay rings. In particular, they show that if \(H\) is the normal semigroup \[ \{(a,b)\in \mathbb{N}_0^2|\,0\leq b/a<\infty\}\cup\{(0,0)\} \] and if \(k\) is a field, then the following are hold:{\parindent=6mm \begin{itemize} \item[(i)] \(k[H]\) is not Cohen-Macaulay in sense of ideals. \item [(ii)] \(k[H]\)is not weak Bourbaki unmixed. \item [(iii)] \(k[H]\) is not Cohen- Macaulay in sense of Hamilton-Marley. \end{itemize}}