Definition 1

Given a ring $\mathbf{A}$, a left module over $\mathbf{A}$ (or left $\mathbf{A}$-*module*) is a set $\mathbf{E}$ with an algebraic structure defined by giving:

(1) a commutative group law on $\mathbf{E}$ (written additively in what follows);

(2) a law of action $(\alpha, x) \mapsto \alpha \top x$, whose domain of operators is the ring $\mathbf{A}$ and which satisfies the following axioms:

Axiom $(\mathbf{M_I})$ means that the external law of a left $\mathbf{A}$-module $\mathbf{E}$ is distributive with respect to addition on $\mathbf{E}$; a module is thus a commutative group with operators.

If in Definition 1, axiom $(\mathbf{M_{III}})$ is replaced by

E with the algebraic structure thus defined is a right module over A or a right A-module.

When speaking of A-modules (left or right), the elements of the ring A are often called scalars.

Usually the external law of composition of a left A-module (resp. right A-module) is written multiplicatively, with the operator written on the left (resp. right); condition (M_{III}) is then written $\alpha(\beta x) = (\alpha\beta)x$, condition (M'_{III}) is written $(x\beta)\alpha = x(\beta\alpha)$.

If $A^0$ denotes the opposite ring to A (I, § 8, no. 3), every right module E over the ring A is a left module over the ring $A^0$. It follows that an exposition can be given of the properties of modules whilst systematically confining attention either to left modules or to right modules; in §§ 1 and 2 we shall in general give this exposition for left modules and when we speak of a module (without specifying which type) we shall mean a left module whose external law will be written multiplicatively. When the ring A is commutative the notions of right module and left module with respect to A are identical.

For all $\alpha \in A$, the mapping $x \mapsto \alpha x$ of an A-module E into itself is called the homothety of ratio $\alpha$ of E (I, § 4, no. 2); by (M_I) a homothety is an endomorphism of the commutative group structure (without operators) on E, but not in general of the module structure on E (I, § 4, no. 2; cf. II, § 1, no. 2 and no. 13). Hence $\alpha 0 = 0$ and $\alpha(-x) = -(\alpha x)$; if $\alpha$ is an invertible element of A, the homothety $x \mapsto \alpha x$ is an automorphism of the commutative group structure (without operators) on E, for the relation $y = \alpha x$ implies by virtue of (M_IV), $x = \alpha^{-1}(\alpha x) = \alpha^{-1}y$.

Similarly, by virtue of (M_{II}), for all $x \in E$, the mapping $\alpha \mapsto \alpha x$ is a homomorphism of the additive group A into the commutative group (without operators) E; hence $0x = 0$ and $(-\alpha)x = -(\alpha x)$; moreover, by (M_IV), for every integer $n \in \mathbf{Z}$, $n.x = (n.1)x$.

When the ring A consists only of the element 0, every A-module E consists only of the element 0, for then $1 = 0$ in A, whence, for all $x \in E$, $x = 1.x = 0.x = 0$.