The authors call a (right distributive) nearring \(N\) a \(P_k(r,m)\) nearring, \(P_k'(r,m)\) nearring, respectively, if \(x^kN=x^rNx^m\), \(Nx^k=x^rNx^m\), for all \(x\in N\), respectively, where \(k,r,m\) are natural numbers. If \(r=m=1\) the respective classes are denoted by \(P_k\) and \(P_k'\). Also considered are \(S_r\) and \(S_r'\) nearrings, they have the property that \(x\in Nx^r\), \(x\in x^rN\) for all \(x\in N\), respectively.
Various properties of such nearrings are derived. Examples are given to show that the concepts do not all coincide.