Modules

A module is an algebraic structure consisting of an abelian group under addition together with a compatible action by a ring. More precisely, if R is a ring and M is an abelian group, M is an R-module if there is a scalar multiplication R x M -> M satisfying (r+s)m = rm + sm, r(m+n) = rm + rn, (rs)m = r(sm), and 1m = m when R has a multiplicative identity. Submodules are subgroups closed under scalar multiplication. Quotients, homomorphisms, and exact sequences parallel those for vector spaces. Examples include abelian groups viewed as Z-modules, and vector spaces as modules over their base field. Free modules generalize bases to modules; modules over polynomial rings or other rings give rise to structure theorems, such as the classification of finitely generated modules over a principal ideal domain, and the decomposition into torsion and free parts.

In software engineering, a module is a self-contained unit that groups related functionality behind a defined interface, promoting encapsulation and reuse. Module systems manage dependencies, namespaces, and loading. Common examples include Python modules (files containing definitions and code), JavaScript modules (ES modules and CommonJS), and Java's module system (JPMS). Good module design emphasizes cohesion within a module and loose coupling between modules, with clear APIs, dependency management, and minimal side effects. Modules can be combined to assemble large systems while enabling testing and maintainability.

In hardware and systems engineering, modules refer to self-contained units that implement a specific function and can be assembled into a larger device or system. This modular approach supports scalability, upgradeability, and parallel development, with interfaces and standards governing communication between modules.