Article
MSC:
05C25,
05E40,
13A30,
13B25,
13C13,
13P25 | MR 4389116 | Zbl 07511563 | DOI: 10.21136/CMJ.2021.0407-20
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Keywords:
strong persistence property; associated prime; cover ideal; symbolic strong persistence property
strong persistence property; associated prime; cover ideal; symbolic strong persistence property
Summary:
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.
References:
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