Definition:Module

Let $$\left({R, +_R, \times_R}\right)$$ be a ring.

Let $$\left({G, +_G}\right)$$ be an abelian group.

A module over $$R$$ or an $$R$$-module is an $R$-algebraic structure with 1 operation $$\left({G, +_G: \circ}\right)_R$$ such that:

$$\forall x, y, \in G, \forall \lambda, \mu \in R$$:
 * VS 1: $$\lambda \circ \left({x +_G y}\right) = \left({\lambda \circ x}\right) +_G \left({\lambda \circ y}\right)$$;
 * VS 2: $$\left({\lambda +_R \mu}\right) \circ x = \left({\lambda \circ x}\right) +_G \left({\mu \circ x}\right)$$;
 * VS 3: $$\left({\lambda \times_R \mu}\right) \circ x = \lambda \circ \left({\mu \circ x}\right)$$.

Note that a module is not an algebraic structure unless $$R$$ and $$G$$ are the same set.

Scalar Ring
In the module $$\left({G, +_G: \circ}\right)_R$$, the ring $$\left({R, +_R, \times_R}\right)$$ is called the scalar ring.

Scalar
In the module $$\left({G, +_G: \circ}\right)_R$$, the elements of $$\left({R, +_R, \times_R}\right)$$ are called scalars.

Zero Scalar
The zero of the scalar ring is called the zero scalar and usually denoted $$0$$, or, if it is necessary to distinguish it from the identity of $$\left({G, +_G}\right)$$, by $$0_R$$.

Unitary Module
Let $$\left({G, +_G: \circ}\right)_R$$ be a module over $$R$$ such that the scalar ring is a ring with unity whose unity is $$1_R$$.

Then $$\left({G, +_G: \circ}\right)_R$$ is a unitary module over $$R$$ or unitary $$R$$-module iff:
 * VS 4: $$\forall x \in G: 1_R \circ x = x $$.