Let $\Delta$ be a pure simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$ and $I_\Delta$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,\ldots,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_\Delta^{(m)}$ ($S/I_\Delta^m$) is clean for all $m\in\mathbb{N}$ and this is equivalent to saying that $S/I_\Delta^{(m)}$ ($S/I_\Delta^m$, 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 $S/I^{(m)}$ is not (pretty) clean for all integer $m\geq3$. If $\dim(\Delta)=1$, we also prove that $S/I_\Delta^{(2)}$ ($S/I_\Delta^2$) is clean if and only if $S/I_\Delta^{(2)}$ ($S/I_\Delta^2$, respectively) is Cohen-Macaulay., Somayeh Bandari, Ali Soleyman Jahan., and Obsahuje bibliografické odkazy