We present a unified framework for representing commutative and Boolean rings through affine and hyperaffine algebraic theories. This yields categorical equivalences between these theories and rings, and leads to a new analysis of certain classes of modules over Boolean rings. The resulting structures naturally capture the algebraic semantics of the if-then-else construct in programming languages.

翻译：我们提出了一个统一框架，通过仿射与超仿射代数理论来表示交换环与布尔环。这建立了这些理论与环之间的范畴等价性，并引出了对布尔环上特定模类的新分析。所得结构自然地捕捉了编程语言中if-then-else结构的代数语义。