Jordan Higher Derivable Mappings on Rings

Let be a ring. We say that a family of maps is a Jordan higher derivable map (without assumption of additivity) on if (the identity map on ) and hold for all and for each . In this paper, we show that every Jordan higher derivable map on a ring under certain assumptions becomes a higher derivation. As its application, we get that every Jordan higher derivable map on Banach algebra is an additive higher derivation. 1. Introduction Let be a ring. An additive mapping is said to be a derivation (resp., Jordan derivation) if (resp., ) holds for any . Note that, in case of -torsion free ring , is equivalent to for all . A map , (without assumption of additivity) is said to be derivable (resp., Jordan derivable) if (resp., ), holds for all . Recall that a ring is prime if implies that either or and is semiprime if implies . Now suppose that is a ring with a nontrivial idempotent . We write . Note that need not have identity element. Put for any . Then by Peirce decomposition of , we have . Throughout this paper, the notation will denote an arbitrary element of and any element can be expressed as . Characterizing the interrelation between the multiplicative and the additive structures of a ring is an interesting topic and has received attention of many mathematicians. It is a well-known result due to Martindale III [1] that every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive. Long ago, Lu [2] proved that “each derivable map on a -torsion free unital prime ring containing a nontrivial idempotent is a derivation.” Motivated by the above result due to Lu [2], in the year 2011, Jing and Lu [3] characterized Jordan derivable maps to a larger class of rings and proved the following theorem. Theorem 1 (see [3, Theorem 1.2]). Let be a ring containing a nontrivial idempotent and satisfying the following conditions for .(i)If for all , then .(ii)If for all , then .(iii)If for all , then . If a mapping satisfies for all , then is additive. In addition, if is -torsion free, then is a Jordan derivation. The concept of derivation was extended to higher derivation by Hasse and Schmidt [4] (see [5, 6] for a historical account and applications). Let be a family of additive mappings and let be the set of nonnegative integers. Following Hasse and Schmidt [4], is said to be a higher derivation (resp., Jordan higher derivation) on if (the identity map on ) and (resp., ) for all and for each . In an attempt to generalize Herstein’s result for higher derivations, Haetinger [7] proved that, on a prime