(An official and an unofficial electronic version are available.)

We define a Carmichael number of order *m* to be a composite
integer *n* such that *n*th-power raising defines an
endomorphism of every **Z**/*n***Z**-algebra that can be
generated as a **Z**/*n***Z**-module by *m* elements.
We give a simple criterion to determine whether a number is a Carmichael
number of order *m*, and we give a heuristic argument (based on an
argument of Erdos for the usual Carmichael numbers) that indicates that
for every *m* there should be infinitely many Carmichael numbers of
order *m*. The argument suggests a method for finding examples of
higher-order Carmichael numbers; we use the method to provide examples
of Carmichael numbers of order 2.