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We deal here with the existence of half-way nonzero divisors for complete intersections and with related properties of their Hilbert functions.

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Abstract. In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an Ngraded ring A of the form A ≥m := ℓ≥m A ℓ and monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n− 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I(λ) := J(λ), where J(λ) = (x λ1

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