An D21: Algebraic structure $M = (X, +, \times)$ is a

**module**over $R$ if and only if

(i) | $G = (X, +)$ is an D23: Abelian group |

(ii) | $\times : R \times G \to G$ is a D274: Left ring action of $R$ on $G$ |

▾ Set of symbols

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Map

▾ Operation

▾ N-operation

▾ Binary operation

▾ Enclosed binary operation

▾ Groupoid

▾ Ringoid

▾ Semiring

▾ Ring

▾ Left ring action

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Map

▾ Operation

▾ N-operation

▾ Binary operation

▾ Enclosed binary operation

▾ Groupoid

▾ Ringoid

▾ Semiring

▾ Ring

▾ Left ring action

Formulation 2

Let $R$ be a D24: Ring.

An D21: Algebraic structure $M = (X, +, \times)$ is a**module** over $R$ if and only if

An D21: Algebraic structure $M = (X, +, \times)$ is a

(i) | $G = (X, +)$ is an D23: Abelian group |

(ii) | $\times : R \times G \to G$ is a D274: Left ring action of $R$ on $G$ |

Also known as

Module over a ring, Left module, Left ring action space