How we can compute the length (length of a composition series) of the Artinian local ring $R=K[x,y]/(x^3,y^3)$ ?
Does the following chain is a saturated chain of ideals of $K[x,y]$ ?
$(x^3,y^3)\subsetneq (x^3,x^2y^2, y^3) \subsetneq (x^3,x^2y,y^3) \subsetneq (x^3,x^2y,xy^2, y^3) \subsetneq (x^3,xy,y^3)\subsetneq (x^3,xy,y^2) \subsetneq (x^3,y) \subsetneq (x^2,y) \subsetneq (x,y).$