Given (commutative) ring extensions  and  at least one of which is minimal, conditions are studied under which at least one of the extensions  and  is minimal. Field-theoretic examples show that the supplementary conditions are needed for such conclusions. Suitable field-theoretic conditions involve concepts such as linear disjointness and normal extensions. For the ring-theoretic setting, the analysis involves the “inert, decomposed, ramified” trichotomy for the types of integral minimal extensions. Many results are given as “duals” of the earlier work on composites of minimal ring extensions due to the first- and fourth-named authors.

minimal ring extension, minimal field extension, crucial maximal ideal, integrality, flat epimorphism, inert, decomposed, ramified, linearly disjoint, overring, prime ideal, total integral closure, pullback, composite.