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Open Access Research

Generalized (σ,τ) higher derivations in prime rings

Mohammad Ashraf* and Almas Khan

Author Affiliations

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India

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SpringerPlus 2012, 1:31  doi:10.1186/2193-1801-1-31

Published: 6 October 2012

Abstract

Let R be a ring and U be a Lie ideal of R. Suppose that σ, τ are endomorphisms of R. A family D = {dn}nNof additive mappings dn:RR is said to be a (σ,τ)- higher derivation of U into R if d0 = IR, the identity map on R and <a onClick="popup('http://www.springerplus.com/content/1/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/1/1/31/mathml/M1">View MathML</a> holds for all a, bU and for each nN. A family F = {fn}nNof additive mappings fn:RR is said to be a generalized (σ,τ)- higher derivation (resp. generalized Jordan (σ,τ)-higher derivation) of U into R if there exists a (σ,τ)- higher derivation D = {dn}nNof U into R such that, f0 = IR and <a onClick="popup('http://www.springerplus.com/content/1/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/1/1/31/mathml/M2">View MathML</a> (resp. <a onClick="popup('http://www.springerplus.com/content/1/1/31/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/1/1/31/mathml/M3">View MathML</a> holds for all a, bU and for each nN. It can be easily observed that every generalized (σ,τ)-higher derivation of U into R is a generalized Jordan (σ,τ)-higher derivation of U into R but not conversely. In the present paper we shall obtain the conditions under which every generalized Jordan (σ,τ)- higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.

Keywords:
Derivation; Higher derivation; Jordan - higher derivation; Lie ideal