On (¿,¿)-Homomorphisms on prime Rings

This book consists of two chapters, in chapter one we reviewed some known definitions and some necessary lemmas which it used in the next chapters. Chapter two consists of four sections: In section one we defined and studied the concepts of (¿,¿)-homomorphism, Jordan (¿,¿)-homomorphism and Jordan triple (¿,¿)- homomorphism on prime ring R . In section two we introduced the concepts of generalized (¿,¿)-homomorphism and generalized Jordan (¿,¿)-homomorphism and we studied these concepts on prime ring R . In section three we defined and studied the concepts of (¿,¿)-higher homomorphism, Jordan (¿,¿)-higher homomorphism and Jordan triple (¿,¿)-higher homomorphism. The main objective of section three is to prove a Jordan (¿,¿)-higher homomorphism from a ring R into 2-torsion free ring R', such that and ¿i ¿i = ¿i ¿i is a Jordan triple (¿,¿)-higher homomorphism. In section four the main objective proved that every generalized Jordan (¿,¿)-higher homomorphism of a ring R into prime ring R' is either generalized (¿,¿)-higher homomorphism or (¿,¿)- higher anti homomorphism.