Rings

A ring is a set equipped with two operations—addition and multiplication—that behave similarly to those in familiar arithmetic. Specifically, a ring allows you to add, subtract, and multiply elements, with addition being commutative and associative, and multiplication associative. Elements may have an additive identity (like zero), and some may have multiplicative identities (like one). Rings can be seen in various mathematical contexts, such as integers, polynomials, or matrices. They provide a framework to study algebraic structures where addition and multiplication interact systematically, forming the foundation for many areas in advanced mathematics.