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Paper IPM / M / 16653  


Abstract:  
Let I be a squarefree monomial ideal in a polynomial ring R=K[x_{1},…, x_{n}] over a field K, \mathfrakm=(x_{1}, …, x_{n}) be the graded maximal ideal of R, and {u_{1}, …, u_{β1(I)}} be a maximal independent set of minimal generators of I such that \mathfrakm\x_{i} ∉ Ass(R/(I\x_{i})^{t}) for all x_{i}  ∏_{i=1}^{β1(I)}u_{i} and some positive integer t, where I\x_{i} denotes the deletion of I at x_{i} and β_{1}(I) denotes the maximum cardinality of an independent set in I.
In this paper, we prove that if \mathfrakm ∈ Ass(R/I^{t}), then t ≥ β_{1}(I)+1. As an application, we verify that under certain conditions?, every unmixed K·· onig ideal is normally torsionfree, and so has the strong persistence property.
In addition, we show that every squarefree transversal polymatroidal ideal is normally torsionfree.
Next, we state some results on the cornerelements of monomial ideals. In particular, we prove that if I is a monomial ideal in a polynomial ring R=K[x_{1}, …, x_{n}] over a field K and z is an I^{t}cornerelement for some positive integer t such that \mathfrakm\x_{i} ∉ Ass(I\x_{i})^{t} for some 1 ≤ i ≤ n, then x_{i} divides z.
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