Symbolic powers and generalized-parametric decomposition of monomial ideals on regular sequences

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 $x^{e_1}_{τ(1)}R+\dots+x^{e_s}_{τ(s)}R$, where $τ$ is a permutation of $\{1,\ldots,d\}$, $s\in\{1,\ldots,d\}$ 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 $k\geq1$, $I^{(k)}=I^k$, 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.