  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 N-graded ring
generated by all homogeneous elements of degree at least m and monomial ideals
in a polynomial ring over a field. For ideals of the first trype we generalize
a recent result of S. 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 obtained by taking integral closures of m-primary ideals
generated by powers of the variables. We obtain classes of normal monomial
ideals and arithmetic critera for deciding when the monomial ideal is not
normal.
