IFP IDEALS IN NEAR-RINGS

A near-ring N is called an IFP near-ring provided that for all a, b, n is an element of N, ab = 0 implies anb = 0. In this study, the IFP condition in a near-ring is extended to the ideals in near-rings. If N I P is an IFP near-ring, where P is an ideal of a near-ring N, then we call P as the IFP-ideal of N. The relations between prime ideals and IFP-ideals are investigated. It is proved that a right permutable or left permutable equiprime near-ring has no non-zero nilpotent elements and then it is established that if N is a right permutable or left permutable finite near-ring, then N is a near-field if and only if N is an equiprime near-ring. Also, attention is drawn to the fact that the concept of IFP-ideal occurs naturally in some near-rings, such as p-near-rings, Boolean near-rings, weakly (right and left) permutable near-rings, left (right) self distributive near-rings, left (right) strongly regular near-rings and left (w-) weakly regular near-rings.