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On products of additive functions (a third approach). (English) Zbl 0777.12001

Summary: Let \(a_ 1,\dots,a_ s:G\to K\) be additive functions from an abelian group \(G\) into a field \(K\) such that \(a_ 1(g)\cdot \dots \cdot a_ s(g)=0\) for all \(g\in G\). If char\((K)=0\), then it is well known that one of the functions \(a_ j\) has to vanish [cf. the author, L. Reich and J. Schwaiger, Aequationes Math. 45, 83-88 (1993; Zbl 0773.39006)]. We give a new proof of this result and show that, if char\((K)>0\), it is only valid under additional assumptions.

MSC:

12E05 Polynomials in general fields (irreducibility, etc.)
15A06 Linear equations (linear algebraic aspects)
39B52 Functional equations for functions with more general domains and/or ranges

Citations:

Zbl 0773.39006
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References:

[1] Bourbaki, N.,Algebra, Part I. Hermann, Paris and Addison-Wesley, Reading, MA, 1974.
[2] Halter-Koch, F., Reich, L. andSchwaiger, J.,On products of additive functions. Aequatione Math., to appear.
[3] Lang, S.,Algebra. Addison-Wesley, Reading, MA, 1965.
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