Jordan Higher Derivable Mappings on Rings

Abstract

LetRbe a ring. We say that a family of mapsD={dn}n∈Nis a Jordan higher derivable map (without assumption of additivity) onRifd0=IR(the identity map onR) anddn(ab+ba)=∑p+q=n‍dp(a)dq(b)+∑p+q=n‍dp(b)dq(a)hold for alla,b∈Rand for eachn∈N. 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.

On k-semi-centralizing maps of generalized matrix algebras

“…(m) = −mU 1 (1) − mU 2 (m) for all m ∈ M.On the similar pattern with G = 0 we find that U 2 (m) = 0 for all m ∈ M. Combining last two expressions we arrive at T 2 (m) = −mU 1(1) for all m ∈ above we can easily find that T 3 (n) = 0, ∆ 3 (n) = 0, V 3 (n) = −U 1 (1)n − U 3 (n)n and U 3 (n) = 0 for all n ∈ N. These lead to V 3 (n) = −U 1 (1)n for all n ∈ N. Now we mention a significant result of this article as follows:Theorem 4 Let S = S(A, M, N, B) be a 2-torsion free generalized matrix algebra over a commutative ring R. Then any k-semi centralizing derivation on S is zero.Proof. Let Φ be a k-semi centralizing derivation on S. Then by Lemma 2, Φ has the following form a) − mn 0 − m 0 n am 0+ T 2 (m) − m 0 b n 0 a − bn 0 + V 3 (n) U 4 (b) + nm 0 + n 0 m , where a ∈ A; b ∈ B; m, m 0 ∈ M; n, n 0 ∈ N and ∆ 1 : A → A, T 2 : M → M, V 3 : N → N, U 4 : B → B are R-linear maps satisfying condition (1) − (4)given in Lemma 2.…”

“…Further, we have [∆ 1 (a), a] k = 0, i.e., ∆ 1 is k-commuting map on A. Further, replacing a by a + 1 in [∆ 1 (a), a] k = 0, we conclude that [∆ 1 (1), a] k = 0 for all a ∈ A and hence ∆ 1 (1) ∈ Z(A) k .…”

“…This implies that T 1 (1) = 0 = V 1 (1). Again on assuming G = 0 0 0 1 in (2), we have T 4 (1) = 0 = V 4 (1).…”

On Jordan Triple Higher Derivable Mappings on Rings

“…In a very recent paper [3], the authors have shown that every Jordan higher derivable map becomes additive with certain assumptions. However, it is still open to find the condition on a ring R under which a Jordan triple higher derivable map turns out to be a Jordan triple higher derivation.…”