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Given a numerical semigroup ring R=k[[S]], an ideal E of S and an odd element bâS, the numerical duplication SâbE is a numerical semigroup, whose associated ring k[[SâbE]] shares many properties with the Nagata's idealization and the amalgamated duplication of R along the monomial ideal I=(teâ£eâE). In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen-Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when grm(I) is Cohen-Macaulay and when grm(ÏR) is a canonical module of grm(R) in terms of numerical semigroup's properties, where ÏR is a canonical module of R.
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