Central ideals and Cartan invariants of symmetric algebras

In this paper, we investigate certain ideals in the center of a symmetric algebra A over an algebraically closed field of characteristic p>0. These ideals include the Higman ideal and the Reynolds ideal. They are closely related to the p-power map on A. We generalize some results concerning these ideals from group algebras to symmetric algebras, and we obtain some new results as well. In case p=2, these ideals detect odd diagonal entries in the Cartan matrix of A. In a sequel to this paper, we will apply our results to group algebras.