Finiteness of the number of irreducible λ-quiddities over a finite commutative and unitary ring

λ-quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion of irreducibility. The main objective of this text is to show that there is a finite number of irreducible λ-quiddities over a finite unitary commutative ring and to obtain in this case an upper bound of the maximal size of them.