Definition:Module

Definition

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 one operation $\left({G, +_G, \circ}\right)_R$ such that:

$\forall x, y \in G, \forall \lambda, \mu \in R$:

$(1): \quad \lambda \circ \left({x +_G y}\right) = \left({\lambda \circ x}\right) +_G \left({\lambda \circ y}\right)$

$(2): \quad \left({\lambda +_R \mu}\right) \circ x = \left({\lambda \circ x}\right) +_G \left({\mu \circ x}\right)$

$(3): \quad \left({\lambda \times_R \mu}\right) \circ x = \lambda \circ \left({\mu \circ x}\right)$

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

Unitary Module

Let $\left({R, +_R, \times_R}\right)$ be a ring with unity whose unity is $1_R$.

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

Let $\left({G, +_G, \circ}\right)_R$ be a module over $R$.

Then $\left({G, +_G, \circ}\right)_R$ is a unitary module over $R$ or unitary $R$-module iff:

$(4): \quad \forall x \in G: 1_R \circ x = x$.

Vector

The elements of $\left({G, +_G}\right)$ are called vectors.

Zero Vector

The identity of $\left({G, +_G}\right)$ is usually denoted $\mathbf 0$, or some variant of this, and called the zero vector.

Note that on occasion it is advantageous to denote the zero vector differently, for example by $e$, or $0_V$ or $0_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.

Also see

• Scalar Ring

• Vector Space

• Results about modules can be found here.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$