Coprimeness of a fractional representation plays various crucial roles in many different contexts, for example, stabilization of a given plant, minimality of a state space representation, etc. It should be noted however that coprimeness depends crucially on the choice of a ring (or algebra) where such a representation is taken, which reflects the choice of a plant, and particular problems that one studies. Such relationships are particularly delicate and interesting when dealing with infinite-dimensional systems. This paper discusses various coprimeness issues for different rings, typically for Hinfinity and pseudorational transfer functions. The former is related to Hinfinity-stabilizability, and the latter to controllability of behaviors. We also give some intricate examples where a seemingly non-coprime factorization indeed turns out to be a coprime factorization over Hinfinity. Some future directions are also indicated.