ABSTRACT: Let R = K[x_1, . . . , x_n] be a polynomial ring over a field K and let be a polymatroidal ideal. This presentation is going to be about some interesting properties and results of polymatroidal ideals. In particular, our focus includes properties of astab (associated prime stability) and dstab (depth stability) of polymatroidal ideals. additionally, we explore Cohen-Macaulay and unmixed polymatroidal ideals. Furthermore, we give some examples such that the astab and dstab are unrelated for such monomial ideals and also we give some known problems and conjectures about this subject.
