We characterize some types of FIP and FCP ring extensions R ⊂ S, where S is not an integral domain and R may not be an integral domain, contrary to a general trend. In each of the sections, S is a product of finitely many rings that are related to R in various ways. Ring extensions of the form Rn → Rp associated to some matrices are also considered. Our tools are minimal ring morphisms and seminormalization, while Artinian conditions on rings are ubiquitous.