We first introduce a new type of dependence on a ring and we study the “algebraic and geometric” properties of a ring containing any element depending on it. We then study the properties of the coordinate ring of an affine variety V such that any subvariety W of V which is a set-theoretic complete intersection is also an algebraic complete intersection. Finally we consider three extensions of UFD rings. These extensions, besides originating new interesting problems, allow us to generalize many results chiefly concerning the cohomological dimension of algebraic varieties, in which the condition UFD is generally assumed.