Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only if $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$,...

Let $G$ be a finite simple graph with the vertex set $V$ and let $I_G$ be its edge ideal in the polynomial ring $S=𝕂\left[V\right]$. We compute the depth and the Castelnuovo-Mumford regularity of $S/I_G$ when $G=G_1\circ G_2$ or $G=G_1*G_2$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_1$, $G_2$ by the $\circ$ operation or $*$ operation, respectively.

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $\left(I^{k+1}{:}_{R}I\right)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $\left(I^{(k+1)}{:}_R{I}^{\left(1\right)}\right)=I^{\left(k\right)}$ for all $k$, where $I^{\left(k\right)}$ 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...