We give easy proofs for some well known facts by using some basic property of cleanness. We show that if (R,m) is a Noetherian local ring and M is a finitely generated almost clean R-module with the property that R/P is Cohen–Macaulay for all P 2 Ass(M), then depth(M) = min{dim(R/P) : p 2 Ass(M)}. Using this fact we show that if M is a finitely generated clean R-module such that R/P is Cohen–Macaulay and dim(M) = dim(R/P) for all minimal prime ideals of M, thenM is Cohen–Macaulay. This implies the well known fact that a pure shellable simplicial complex is Cohen–Macaulay.
