In this paper, we introduce the concept of k-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of k-clean ideals, we show that a (d−1)-dimensional simplicial complex is k-decomposable if and only if its Stanley-Reisner ideal is k-clean, where k ≤ d − 1. We prove that the classes of monomial ideals like Cohen-Macaulay ideals of codimension 2, monomial ideals of forest type without embedded prime ideal and symbolic powers of StanleyReisner ideals of matroid complexes are k-clean for all k ≥ 0.
