Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Affine semigroup rings that satisfy Serre's condition Rk by Marie VITULLI, University of Oregon Abstract: We will characterize those affine semigroup rings R = K[S ] over an arbitrary field K that satisfy condition Rk of Serre for k < dim(K[S]). Our characterization is in terms of the face lattice of the positive cone pos(S ) of S. After introducing our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of nonnormal monomial ideals whose Rees algebras satisfy Rk.