Let $K$ be a field and $S=K[x_1,\ldots , x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots , u_r$ be monomials in $S$. We prove that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots , u_r)$ is pretty clean. Also, we show that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then Stanley's conjecture is true for $S/I$ if and only if it is true for $S/(I,u_1, \ldots , u_r)$. Finally, we prove that if $u_1,\ldots , u_r$ is a minimal set of generators for $I$ which form either a $d$-sequence, proper sequence or strong $s$-sequence (with respect to the reverse lexicographic order), then $S/I$ is pretty clean.
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