Module theory is a branch of abstract algebra that generalizes the concept of vector spaces. In this theory, a module is a mathematical structure consisting of a set equipped with an operation that combines elements of the set with elements from a ring, similar to how vectors combine with scalars in vector spaces. This allows for the study of linear algebra concepts in a broader context. Modules can be classified based on their properties, such as being free, projective, or injective. The study of modules over rings, like Z (the integers) or R (the real numbers), helps mathematicians understand the relationships between algebraic structures and their applications in various fields, including geometry and number theory.