 is a finitely generated almost clean
 is a finitely generated almost clean  -module with the property that
-module with the property that  is Cohen-Macaulay for all 
P ∈ Ass(M), then depth(M)=min{dim(R/P) : p ∈ Ass(M)}. Using this fact we show that if
 is Cohen-Macaulay for all 
P ∈ Ass(M), then depth(M)=min{dim(R/P) : p ∈ Ass(M)}. Using this fact we show that if  is a finitely generated clean
 is a finitely generated clean  -module such that
-module such that  is Cohen-Macaulay and
 is Cohen-Macaulay and 
 for all minimal prime ideals of
 for all minimal prime ideals of  , then
, then  is Cohen-Macaulay. This implies the well known fact that a pure shellable simplicial complex is  Cohen-Macaulay.
 is Cohen-Macaulay. This implies the well known fact that a pure shellable simplicial complex is  Cohen-Macaulay.
Key Words: Prime filtration, Shellable simplicial complex, Monomial ideals, Clean and pretty clean modules.
2000 Mathematics Subject Classification: Primary: 13C13
Secondary: 13F55, 13F20.
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