The Alienation Phenomenon and Associative Rational Operations

• Autorzy: Roman Ger
• Języki publikacji: EN

Abstrakty

EN

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. (original abstract)

Słowa kluczowe

EN

Mathematics

PL

Matematyka

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2013
• Tom: 27
• Strony: 75--88

Twórcy

autor

Roman Ger

• University of Silesia in Katowice, Poland

Bibliografia

• Chéritat A., Fractions rationnelles associatives et corps quadratiques, Revue des Mathématiques de l'Enseignement Supérieur 109 (1998-1999), 1025-1040.
• Dhombres J., Relations de dépendance entre équations fonctionnelles de Cauchy, Aequationes Math. 35 (1988), 186-212.
• Ger R., On an equation of ring homomorphisms, Publ. Math. Debrecen 52 (1998), 397-417.
• Ger R., Ring homomorphisms equation revisited, Rocznik Nauk.-Dydakt. Prace Mat. 17 (2000), 101-115.
• Ger R., Reich L., A generalized ring homomorphisms equation, Monatsh. Math. 159 (2000), 225-233.