On the derivations of semigroup rings (Q1329615)

Let \(k\) be a field of characteristic zero, \(S\) a submonoid of \(\mathbb{Z}^ n_ +\) \((n \geq 2)\), and \(R = k[S]\) the monoid ring. Thus \(R\) is a subring of \(R' = k[t_ 1,\dots,t_ n]\) generated by monomials over \(k\). In this paper, the authors study, \(\text{Der}_ k(R)\), the \(R\)-module of \(k\)-derivations of \(R\) for certain submonoids \(S\) of \(\mathbb{Z}_ +^ n\). For such an \(R\), they show that each \(D \in \text{Der}_ k (R)\) has the form \(D = \sum_ j P_{1j} \partial/ \partial t_ 1 + \dots + \sum_ j P_{nj} \partial/ \partial t_ n\), where for each \(i\), the \(P_{ij}\)'s are monomials of different degrees in \(R'\). They also find generators for the submodule \(E = \{D \in \text{Der}_ k(R) \mid P_{ij} \partial /\partial t_ i \in \text{Der}_ k(R)\text{ for each }i,\;j\}\) in certain cases when \(R\) is Cohen-Macaulay and study the set \(F = \{D \in \text{Der}_ k(R) \mid P_{ij} \partial /\partial t_ i \not\in \text{Der}_ k(R)\text{ for each }i,\;j\}\).