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Title: The alienation phenomenon and associative rational operations
Authors: Ger, Roman
Keywords: Alienation phenomenon; Additive and multiplicative mappings
Issue Date: 2013
Citation: Annales Mathematicae Silesianae, Nr 27 (2013), s. 75-88
Abstract: The alienation phenomenon of ring homomorphisms may briefly be described as follows: under some reasonable assumptions, a map f between two rings satisfies the functional equation f(x+y)+f(xy)=f(x)+f(y)+f(x)f(y)(∗) if and only if f is both additive and multiplicative. Although this fact is surprising for itself it turns out that that kind of alienation has also deeper roots. Namely, observe that the right hand side of equation (∗) is of the form Q(f(x),f(y)) with the map Q(u,v)=u+v+uv being a special rational associative operation. This gives rise to the following question: given an abstract rational associative operation Q does the equation f(x+y)+f(xy)=Q(f(x),f(y)) force f to be a ring homomorphism (with the target ring being a field)? "Plainly, in general, that is not the case. Nevertheless, the 2-homogeneity of f happens to be a necessary and sufficient condition for that effect provided that the range of f is large enough.
ISSN: 0860-2107
Appears in Collections:Artykuły (WNŚiT)

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