\chapter{Modules}

Readers are familiar with linear spaces as a field and set where scalar multiplication and vector addition are defined. Typically linear algebra deals with fields on $\mathbb{R}$ or $\mathbb{C}$.

Generalizing the idea of a linear space to be over any ring leads to the idea of a \emph{module}.







Module theory permits the following short and sweet definition for a linear space.

\begin{definition}[Linear space (Module theory)]
A \emph{linear space} is a module where the ring is a field.
\end{definition}