Commutative algebra is a branch of mathematics that studies the properties and structures of commutative rings, which are algebraic systems where the multiplication operation is commutative. This means that the order in which two elements are multiplied does not affect the result. Commutative algebra provides tools to understand polynomial equations and their solutions, making it essential in various areas of mathematics, including algebraic geometry and number theory. One of the key concepts in commutative algebra is the idea of an ideal, which is a special subset of a ring that allows for the construction of quotient rings. These ideals help in analyzing the structure of rings and their homomorphisms. The study of modules over rings is also significant, as it generalizes the notion of vector spaces and provides deeper insights into the relationships between different algebraic structures.