Abstract | Let A be a graded k-algebra and M be a finitely generated A-module. The Poincaré series P M A ( z ) is the formal power series ∑ i dim k Tor i A ( k , M ) z i . We study the Poincaré series of Der k k [ S ] , the module of derivations of a numerical semigroup ring k [ S ] , and we relate it to the Poincaré series of k over k [ S ] and to the type of S. We then use this in order to determine the Poincaré series of Der k k [ S ] or, at least, its rationality, for some classes of examples. We finally give an example of a non-rational P Der k k [ S ] k [ S ] ( z ) . |