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# On Stanley Depths of Certain Monomial Factor Algebras

Published:2015-02-19
Printed: Jun 2015
• Zhongming Tang,
Department of Mathematics, Suzhou University, Suzhou 215006, PR~China
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## Abstract

Let $S=K[x_1,\ldots,x_n]$ 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 ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {size}}_S(I)$. When $I$ is squarefree and ${\operatorname {bigsize}}_S(I)\leq 2$, the Stanley conjecture holds for $S/I$, i.e., ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {depth}}_S(S/I)$.
 Keywords: monomial ideal, size, Stanley depth
 MSC Classifications: 13F20 - Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 13C15 - Dimension theory, depth, related rings (catenary, etc.)

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