Decidability in Algebraic Geometry

The central theme of our investigation is the concept of Decidability in Algebra/Algebraic Geometry. To the best of our knowledge this seems to be novel in the sense that there is no work known to isolate or to focus on the concept of Decidability in the context of Commutative Algebra. Decidability is more restrictive than Grothendieck's concept of formally unramified, but weaker than the concept of étale. In this article we study these relationships by characterizing Decidability for ring-extensions of essentially finite type. In the absence of essential finiteness we can only show, at present, that a separable algebraic extension of fields is indeed Decidable.