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- W2901326722 abstract "Let $R$ be a commutative Noetherian ring and let ${bf x} :=x_1,ldots,x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to ${bf x}$ if it is generated by a set of monomials $x_1^{e_1}ldots x_d^{e_d}$. It is shown that, if ${bf x}R$ is a prime ideal of $R$, then each monomial ideal $I$ has a canonical and unique decomposition as an irredundant finite intersection of primary ideals of the form $x^{e_1}_{tau(1)}R+dots+x^{e_s}_{tau(s)}R$, where $tau$ is a permutation of ${1,ldots,d}$, $sin{1,ldots,d}$ and ${e_1},ldots,{e_s}$ are the positive integers. This generalizes and provides a short proof of the main results of cite{HMRS, HRS}. Also, we show that for every integer $kgeq1$, $I^{(k)}=I^k$, if and only if $Ass_R R/{I^k }subseteq Ass_R R/{I}$, whenever $I$ is a squarefree monomial ideal, where $I^{(k)}$ is the $k$th symbolic power of $I$. Moreover, in this circumstance it is shown that all powers of $I$ are integrally closed." @default.
- W2901326722 created "2018-11-29" @default.
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- W2901326722 date "2018-11-16" @default.
- W2901326722 modified "2023-09-27" @default.
- W2901326722 title "Symbolic powers and generalized-parametric decomposition of monomial ideals on regular sequences" @default.
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