Characterizing ring derivations of all orders via functional equations: results and open problems. (English) Zbl 1358.39010
The authors focus on a class of functional equations for additive functions that have derivations (of various orders) as solutions. They solve functional equations that characterize derivations among all the additive maps on a given ring. In the process, they provide a unifying framework for the treatment of equations of the form
\[
\sum_{k=1}^nx^{pk}f_k(x^{qk})=0
\]
for additive maps \(f_k\) and integers \(p_k,q_k\) \((1\leq k\leq n)\).
Also, they propose some problems for further research. These results are stated for functions on integral domains of characteristic zero. For this consideration, they restrict their attention to fields of characteristic zero. The authors introduce some examples of ring derivations.
It is worth to mention that these results could be formulated for integral domains or fields of sufficiently large characteristic as well.
Also, they propose some problems for further research. These results are stated for functions on integral domains of characteristic zero. For this consideration, they restrict their attention to fields of characteristic zero. The authors introduce some examples of ring derivations.
It is worth to mention that these results could be formulated for integral domains or fields of sufficiently large characteristic as well.
Reviewer: Nasrin Eghbali (Ardabil)
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 13N15 | Derivations and commutative rings |
| 16W25 | Derivations, actions of Lie algebras |
Keywords:
ring derivation; derivation of higher order; additive map; homogeneous function; integral domain; functional equationsReferences:
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