Algebraic structure of Schur rings

Schur ring (S-ring) is known to have applications on group theory and combinatorial design theory (see [1],[5]). In this paper, we study the general structure of Schur rings. Schur subrings, normal Schur subrings, quotient S-rings, S-ring homomorphism and direct product of two S-rings are introduced. As in group theory, three isomorphism theorems of S-rings are shown. We also show that the set of all normal Schur subrings of a given S-ring ordered by inclusion is a modular lattice. Hence it satisfies the Jordan-Hölder-Dedekind theorem. However some of our results are found different from the structure of groups. For instance, the kernel of an S-ring homomorphism may not be normal; the direct product of two Schur subrings of an S ring may not be its Schur subring.