We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences with respect to the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semidegenerate congruence–modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive with respect to arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence–modular varieties.