On Stanley Depths of Certain Monomial Factor Algebras

Let $S\,=\,K\left[ {{x}_{1}},\,\ldots \,,\,{{x}_{n}} \right]$ be the polynomial ring in $n$ -variables over a field $K$ and $I$ a monomial ideal of $S$ . According to one standard primary decomposition of $I$ , we get a Stanley decomposition of the monomial factor algebra $S/I$ . Using this Stanley decomposition, one can estimate the Stanley depth of $S/I$ . It is proved that $\text{sdept}{{\text{h}}_{s}}\left( S/I \right)\,\ge \,\text{siz}{{\text{e}}_{S}}\left( I \right)$ . When $I$ is squarefree and $\text{bigsiz}{{\text{e}}_{S}}\left( I \right)\,\le \,2$ , the Stanley conjecture holds for $S/I$ , i.e., $\text{sdept}{{\text{h}}_{S}}\left( S/I \right)\ge \text{dept}{{\text{h}}_{S}}\left( S/I \right)$ .