A proof is given of the Ferrand-Olivier result that if kis a field, then a nonzero k-algebra B is (when viewed as a ring extension of kby means of the injective structural ring homomorphism  a minimal ring extension of k if and only if B is k-algebra isomorphic to (exactly one of) a minimal field extension of k,  or  where X is transcendental over k. A survey is included of some consequences of that result and of studies concerning the Ferrand-Olivier concept of a crucial maximal ideal, especially in regard to the FCP and FIP properties of ring extensions.

minimal ring extension, crucial maximal ideal, integrality, overring, integral domain, ramified extension, inert extension, decomposed extension, flat epimorphism, FCP, FIP, normal pair.