We study some basic property of cleanness. We show that if R is a Notherian ring and M is an almost clean R-module with the property that R/P is Cohen–Macaulay for any PÎAss(M), then depth (M)=min{dim(R/P) : pÎAss(M)}. Using this fact show that if M is a clean R-module and all minimal prime ideals of M are Cohen–Macaulay and have equal height, then M is Cohen–Macaulay. We also discuss the relation between cleanness and shalleblity and give a simple proof for a theorem of Dress. Finally we give an easy proof for the well known fact that a pure shellable simplicial complex is Cohen–Macaulay.
