Stanley Depth Of Certain Monomial Ideals

In this paper, we mainly do three parts of work. Let K be a field and S be the polynomial ring over K in n variables. Let m be the maximal ideal of S.First, let I S be a monomial ideal and I = s i=1Qibe an irredundant decomposition of I, where each Qiis Pi- primary. We first show that the Stanley conjecture holds for S/I and I if Pi = m, i ∈ [s], P1+ P2= m and Q3∩ · · · ∩ Qs Q1+ Q2. Then we show that the Stanley conjecture holds for I if Pi = sj=1Pj, i ∈ [s] and Qi1∩ · · · ∩ Qit Qj1+ · · · + Qjs-t, 1 ≤ t ≤ s- 2, i1< · · · < it∈ [s], j1< · · · < js-t∈ [s] \ {i1,..., it}.Secondly, let Q1, Q2, Q3, Q4 be four non-zero irreducible monomial ideals of S, where each Qiis Pi- primary. We first show that the Stanley conjecture holds for S/I and I if G(Pi) ∩ G(Pj) = {xα1,..., xαl}, i = j and {xα1,..., xαl} G(Pi), i ∈ [4]. Then we show that the Stanley conjecture holds for S/I if G(Pi) ∩ G(Pj) = , i = j.Thirdly, we give an upper bound for the Stanley depth of I according to the relationship among the associated prime ideals of I, where I is a monomial ideal of S.