The strong persistence property and symbolic strong persistence property

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

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 1, page 209-237
  • ISSN: 0011-4642

Abstract

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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 : R I ) = I k for all k , and I has the symbolic strong persistence property if and only if ( I ( k + 1 ) : 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.

How to cite

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Nasernejad, Mehrdad, et al. "The strong persistence property and symbolic strong persistence property." Czechoslovak Mathematical Journal 72.1 (2022): 209-237. <http://eudml.org/doc/297679>.

@article{Nasernejad2022,
abstract = {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.},
author = {Nasernejad, Mehrdad, Khashyarmanesh, Kazem, Roberts, Leslie G., Toledo, Jonathan},
journal = {Czechoslovak Mathematical Journal},
keywords = {strong persistence property; associated prime; cover ideal; symbolic strong persistence property},
language = {eng},
number = {1},
pages = {209-237},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The strong persistence property and symbolic strong persistence property},
url = {http://eudml.org/doc/297679},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Nasernejad, Mehrdad
AU - Khashyarmanesh, Kazem
AU - Roberts, Leslie G.
AU - Toledo, Jonathan
TI - The strong persistence property and symbolic strong persistence property
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 1
SP - 209
EP - 237
AB - 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.
LA - eng
KW - strong persistence property; associated prime; cover ideal; symbolic strong persistence property
UR - http://eudml.org/doc/297679
ER -

References

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