The strong persistence property and symbolic strong persistence property

Nasernejad, Mehrdad ; Khashyarmanesh, Kazem ; Roberts, Leslie G. ; Toledo, Jonathan

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 209-237

Abstract (VO)

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $(I^{k+1}\colon _R I)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $(I^{(k+1)}\colon _R I^{(1)})=I^{(k)}$ for all $k$, where $I^{(k)}$ denotes the $k$th symbolic power of $I$. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial ideal has the symbolic strong persistence property.

Détail

DOI : 10.21136/CMJ.2021.0407-20

Classification : 05C25, 05E40, 13A30, 13B25, 13C13, 13P25
Keywords: strong persistence property; associated prime; cover ideal; symbolic strong persistence property

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