d $R$-module. A basic fact in commutative algebra (see [@M Theorem 6.4]) says that there exists a finite filtration $$\mathcal{F}\: 0=M_0\subset M_1\subset \cdots \subset M_{r-1}
\subset M_r=M$$ with cyclic quotients $M_i/M_{i-1}\iso R/P_i$ and $P_i\in \Supp(M)$. We call any such filtration of $M$ a [*prime filtration*]{}. The set of prime ideals $\{P_1,\ldots, P_r\}$ which define the cyclic quotients of $\mathcal{F}$ will be denoted by $\Supp(\mathcal{F})$. Another basic fact [@M Theorem 6.5] says that $$\Ass(M)\subset \Supp(\mathcal{F})\subset \Supp(M).$$ Let $\Min(M)$ denote the set of minimal prime ideals. Dress [@D] calls a prime filtration ${\mathcal F}$ of $M$ [*clean*]{}, if $\Supp(\mathcal{F})\subset\Min(M)$. The module $M$ is called [*clean*]{}, if $M$ admits a clean filtration. It is clear that for a clean filtration $\mathcal
F$ of $M$ one has $$\Min(M)=\Ass(M)=\Supp({\mathcal F}).$$

Cleanness is the algebraic counterpart of shellability for simplicial complexes. Indeed, let $\Delta$ be a simplicial complex and $K$ a field. Dress [@D] showed that $\Delta$ is (non-pure) shellable in the sense of Björner and Wachs [@BW], if and only if the Stanley-Reisner ring $K[\Delta]$ is clean.

On the other hand Stanley [@St] showed that if $\Delta $ is shellable, then $K[\Delta]$ is sequentially Cohen-Macaulay. In this paper we show more generally that any clean module over a Cohen-Macaulay ring which admits a canonical module is sequentially Cohen-Macaulay if all factors in the clean filtration are Cohen-Macaulay. In fact, we prove this result (Theorem \[sequentially\]) for an even larger class of modules which we call pretty clean. These modules are defined by the property that they have a prime filtration as above, and such that for all $i<j$ for which $P_i \subset P_j$ it follows that $P_i=P_j$.

We now describe the content of this paper in more detail. In Section 1 we recall the concept of dimension filtrations introduced by Schenzel [@Sc], and note (Proposition \[characterization\]) that the dimension filt