Semigroup structure of sets of solutions to equation \(X^m=X^s\)

Abstract. We describe the semigroup and group structure of the set of solutions to equation \(X^m=X^s\) over the multiplicative semigroups of factor rings of residually finite commutative rings and of residually finite commutative PID's. The analysis is done in terms of the structure of maximal unipotent subsemigroups and subgroups of semigroups of the corresponding rings. In case of residually finite PID's we employ the available idempotents analysis of the Euler–Fermat Theorem in these rings used to determine minimal positive integers \(\nu\) and \(\mu\) such that for all elements \(x\) of these rings one has \(x^{\kappa+\delta}=x^\kappa\). In particular, the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to \(x^n=x\pmod{n}\) connected with the message space of RSA cryptosystems and Fermat pseudoprimes.