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| Paper IPM / M / 16653 |
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| Abstract: | |
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Let I be a square-free monomial ideal in a polynomial ring R=K[x1,…, xn] over a field K, \mathfrakm=(x1, …, xn) be the graded maximal ideal of R, and {u1, …, uβ1(I)} be a maximal independent set of minimal generators of I such that \mathfrakm\xi ∉ Ass(R/(I\xi)t) for all xi | ∏i=1β1(I)ui and some positive integer t, where I\xi denotes the deletion of I at xi and β1(I) denotes the maximum cardinality of an independent set in I.
In this paper, we prove that if \mathfrakm ∈ Ass(R/It), then t ≥ β1(I)+1. As an application, we verify that under certain conditions?, every unmixed K·· onig ideal is normally torsion-free, and so has the strong persistence property.
In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free.
Next, we state some results on the corner-elements of monomial ideals. In particular, we prove that if I is a monomial ideal in a polynomial ring R=K[x1, …, xn] over a field K and z is an It-corner-element for some positive integer t such that \mathfrakm\xi ∉ Ass(I\xi)t for some 1 ≤ i ≤ n, then xi divides z.
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