Properties of Boolean Rings Additivity and Commutativity

A Boolean ring is any ring where for any element $x$, $x^2 = x$. Prove the following: 1. For all elements $x$ in a Boolean ring, $x + x = 0$. 2. Every Boolean ring is commutative, i.e., for all elements $x$ and $y$ in a Boolean ring, $xy = yx$.

Boolean rings are an interesting area of study in abstract algebra and provide intriguing properties that are not immediately intuitive. The main property defining a Boolean ring is that every element is idempotent, meaning when squared, it equals itself. This property leads to several remarkable results that we explore in this problem. The first result, "x + x = 0 for all elements x in a Boolean ring," demonstrates a property akin to involution, suggesting that adding an element to itself cancels out to the zero element, similar to the additive inverses in familiar number systems. This result utilizes the idempotent nature of the elements extensively and relies on the ring's additive identity properties. The second part examines the commutative nature of Boolean rings. Finding that all Boolean rings are commutative means that the order in which you multiply elements does not matter, which aligns them with a broader class of algebraic structures known as commutative rings. Proving this requires leveraging the idempotent property along with the previously shown additivity result. Although this commutativity is expected in many familiar algebraic settings, it is striking here because Boolean rings can behave distinctively in other aspects, such as not necessarily containing multiplicative inverses. This exploration of Boolean rings highlights the elegance of abstract algebra, where simple conditions on elements lead to broad and powerful results, and the study of such structures illuminates the underpinnings of other mathematical systems.