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Abstract. In this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.

In this paper, we show that for any Schur ring S over a cyclic group G, if every subgroup is an S-subgroup, then S is either a wedge product of Schur rings over smaller cyclic groups, or every S-principal subset is an orbit of an element under a Ž fixed subgroup of Aut G. With an earlier result proved by us on Schur rings over . cyclic groups, preprint , we are able to determine all possible structures of Schur rings over a cyclic group. ᮊ

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