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). In a commutative ring, there is also an additive identity (usually denoted as 0) and a multiplicative identity (usually denoted as 1). Elements in the ring can be added or multiplied together, and the results will also belong to the same set. Examples of commutative rings include the set of integers \mathbb{Z} and the set of polynomials with real coefficients \mathbb{R[x]}.