Finiteness of the number of irreducible $\lambda$-quiddities over a finite commutative and unitary ring

A $\lambda$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises 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 demonstrate that there is a finite number of irreducible $\lambda$-quiddities over a finite unitary commutative ring and to obtain in this case an upper bound for their maximal size.