Unifying Boolean and Algebraic Descriptive Complexity

We introduce ultrarings, which simultaneously generalize commutative rings and Boolean lextensive categories. As such, they allow to blend together standard algebraic notions (from commutative algebra) and logical notions (from categorical logic), providing a unifying descriptive framework in which complexity classes over arbitrary rings (as in the Blum, Schub, Smale model) and usual, Boolean complexity classes may be captured in a uniform way.