Simple Morphisms in Algebraic Geometry The Zariski topology the Jacobian criterion and examples of simple algebras over a field k The Kahler 1 differentials Every k algebra a which is essentially of finite type over k and simple is a regular local ring Brief discussion of unramified and étale homomorphisms Some corollaries to Theorem 3.5 Fitting ideals Proof of the Jacobian criterion and some characterizations of simple k algebras and A algebras Characterizations of simple A algebras in terms of étale homomorphisms; invariance of the property of being a simple algebra under composition and change of base Descent of simple homomorphisms and removal of all noetherian assumptions in Chapter 7 and Chapter 8 Simple morphisms of preschemes and translation of previous theorems into the language of preschemes.