A commutative ring is a mathematical structure consisting of a set equipped with two operations: addition and multiplication. In a commutative ring, both operations must satisfy certain properties, such as associativity and distributivity. Additionally, multiplication is commutative, meaning that the order of the elements does not affect the result (e.g., a \cdot b = b \cdot a). An important aspect of commutative rings is the presence of an additive identity (usually denoted as 0) and a multiplicative identity (usually denoted as 1). Examples of commutative rings include the set of integers, denoted as \mathbb{Z}, and the set of polynomials with real coefficients, denoted as \mathbb{R[x]}. These structures are fundamental in various areas of mathematics, including algebra and number theory.