In page 297 of Pilz[4] a right near-ring N is called a near-ring if xN=xNx(Nx=xNx) for all x in N . Szasz, Frence, in [6] calls a ring N , with the property xN=xNx for all x in N , a -ring. We shall, in this paper, refer to a near-ring N with the property xN=xNx for all x in N , a near-ring. Motivated by these concepts we introduce P k and P k ' near-rings (Definition 2.1). We further generalize these concepts by introducing P k (r,m) and P k '(r,m) near-rings (Definition 3.1). We discuss the properties of all these newly introduced structures in detail. We also obtain complete charcterisations and structure theorems for such near rings.