Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?

Given $n\in\mathbb{Z}$ , its arithmetic derivative n' is defined as follows: (i) 0'=1'=(-1)'=0. (ii) If $n=up_1\cdots p_k$ , where $u=\pm 1$ and $p_1,\dots,p_k$ are primes (some of them possibly equal), then

$\begin{displaymath}n'=n\sum_{j=1}^k\frac{1}{p_j}=u\sum_{j=1}^kp_1\cdots p_{j-1}p_{j+1}\cdots p_k. \end{displaymath}$

An analogous definition can be given in any unique factorization domain. What about the converse? Can the arithmetic derivative be (well-)defined on a non-unique factorization domain? In the general case, this remains to be seen, but we answer the question negatively for the integers of certain quadratic fields. We also give a sufficient condition under which the answer is negative.

Can the Arithmetic Derivative be Defined on a Non-Unique Factorization Domain?

Pentti Haukkanen, Mika Mattila, and Jorma K. Merikoski School of Information Sciences FI-33014 University of Tampere Finland Timo Tossavainen School of Applied Educational Science and Teacher Education University of Eastern Finland P.O.Box 86, FI-57101 Savonlinna Finland

Pentti Haukkanen, Mika Mattila, and Jorma K. Merikoski
School of Information Sciences
FI-33014 University of Tampere
Finland

Timo Tossavainen
School of Applied Educational Science and Teacher Education
University of Eastern Finland
P.O.Box 86, FI-57101 Savonlinna
Finland