Module over a ring

Definition

Let $R$ be a ring with identity element $1$. A left module over $R$ is an abelian group $(M, +)$ together with a map $R \times M \to M$, $(r, x) \mapsto rx$, such that, for all $r, s \in R$ and $x, y \in M$:

1. $r(x + y) = rx + ry$.
2. $(r + s)x = rx + sx$.
3. $(rs)x = r(sx)$.
4. $1x = x$.

Remarks

If $R$ is a field, then an $R$-module is the same as a vector space over $R$. Each abelian group is a module over the ring $\Z$, with $nx = x + \cdots + x$ ($n$ terms) for $n > 0$. The ring $R$ is a module over itself, and $R^n$ is a module with the operations on coordinates. Right modules use a map $M \times R \to M$; over a commutative ring, the two notions are the same.

Explanations

L1 human

A module is a vector space in which the scalars come from a ring instead of a field. We cannot divide by scalars in general, so many facts about vector spaces fail. For example, a module need not have a basis.