We introduce a family of subrings of the Chow rings of the moduli spaces of r-spin structures called the tautological rings with effective spin structures. These rings are an enrichment of the standard tautological rings with the classes of loci of effec- tive spin structures. We will describe the product in these rings and explain how this presentation of the ring structure together with conjectures of Janda, Pandharipande, Pixton and Zvonkine should allow to compute intersections of Wittens classes.