Higher-order Carmichael numbers

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 ${\mathbf Z}/n{\mathbf Z}$-algebra that can be generated as a ${\mathbf Z}/n{\mathbf Z}$-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 $2$.