On some alternative functional equations

Abstract

(cf. [1], [2], [4}-[8]). The most general results in this direction were claimed by P. Fischer [1]. They read as follows ( yX denotes here the class of all functions [ : X ~ Y): (A) If (X, +) is a semigroup and (Y, +, .) is an integral domain, then in the class ,,ix the functional equations (1) and (2) are equivalent. (B) If (X, +) is a semigroup and (Y, +, .) is a commutative ring, then in the class yX the functional equations (1) and (2) are equivalent if and only if Y does not contain genuine nilpotent elements. However, Fischer's proof of (A) is erroneous and the results are false. In this situation, the most general correct results concerning the equivalence of (1) and (2) are due to E. Vincze [7] and to H. Swiatak and M. Hossztl [5]. The first author proved the equivalence of (1) and (2) in the class yx , where (X, +) is an arbitrary semigroup, and (Y, +, .) is a suitable number field (and hence, in particular, is of characteristic zero). H. Swiatak and M. Hosszfi have dealt with a more general equation. Their result applied to equation (1) yields the equivalence of (1) and (2) in the class yX, where (X, +) is an arbitrary group and (Y, +, .) is an integral domain of characteristic zero; however, the existence of the multiplicative unit in Y and the commutativity of the + operation in Y were not needed.