Absolutely integral homomorphisms

Absolutely integral algebras over a field are the subject of a chapter in Field Theory. They are also known as geometrically integral algebras in Algebraic Geometry. We generalize this concept to arbitrary commutative algebras. Our first main result is that a ring homomorphism is absolutely integral if and only if it becomes flat after an integral base change and its fibers are geometrically integral. These homomorphisms are characterized with the help of absolute integral closedness property and do have some transcendence properties. They generally preserve normality, regularity, and the complete intersection property.