Abstract
We introduce the concept of Stanley decompositions in the localized polynomial ring S (f) where f is a product of variables, and we show that the Stanley depth does not decrease upon localization. Furthermore it is shown that for monomial ideals J subset of i subset of S(f) the number of maximal Stanley spaces in a Stanley decomposition of I/J is an invariant of I/J. For the proof of this result we introduce Hilbert series for Z(n) -graded K-vector spaces.