m-CANONICAL IDEALS IN SEMIGROUPS

m-CANONICAL IDEALS IN SEMIGROUPS

For a grading monoid S, we prove that (1) if (S, M) is a valuation semigroup, then M is an m-canonical ideal, that is, an ideal M such that M : (M:J)=J for every ideal J of S. (2) if S is an integrally closed semigroup and S has a principal m-canonical ideal, then S is a valuation semigroup, and (3) if S is a completely integrally closed and S has an m-canonical ideal I, then every ideal of S is I-invertible, that is, J+(I+J)=I for every ideal J of S.

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