1. The term "module" in mathematics commonly refers to a generalization of vector spaces where scalars come from a ring instead of a field. 2. A module over a ring $R$ is an additive abelian group $M$ equipped with an operation $R \times M \to M$ satisfying compatibility conditions similar to vector spaces. 3. Key properties defining an $R$-module include: \- Distributivity of scalar multiplication over module addition: $r(m+n) = rm + rn$ \- Distributivity over ring addition: $(r+s)m = rm + sm$ \- Associativity: $(rs)m = r(sm)$ \- Identity: $1_R m = m$ if $1_R$ is the multiplicative identity in $R$ 4. Examples of modules include vector spaces (when $R$ is a field), abelian groups as modules over the integers, and many structures in abstract algebra. 5. Modules are useful because they allow linear-algebra-like reasoning in more general algebraic contexts. Final answer: A module is a mathematical structure generalizing vector spaces by allowing scalars from rings, equipped with operations satisfying certain axioms similar to those in vector spaces.