module

Generalization of vector space, with scalars in a ring instead of a field

EntityQ18848

Also known as module over a ring

Subclass of: algebraic structure

Overview

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In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.

Overview

All data

44 statements · 14% with external sources · numbers in brackets are citations

Defining formula: \begin{aligned}m+n&=n+m &&(m,n\in M)\\ (m+n)+p&=m+(n+p)&&(m,n,p\in M) \\ m+0_M &= m&&(m\in M) \\ (-m)+m &= 0_M&&(m\in M) \\ r(sm) &= (rs)m &&(m\in M,\;r,s\in R)\\ 1_Rm &=m&&(m\in M) \\ r(m+n)&=rm+rn&&(m,n\in M,\;r\in R) \\ (r+s)m &= rm+sm&&(m\in M,\;r,s\in R)\end{aligned}
Different from: absolute value
Members / parts: abelian group
In defining formula: m+n=n+m\qquad(m,n\in M) symbol represents: commutative property
\begin{aligned}(m+n)+p&=m+(n+p)&&(m,n,p\in M) \\r(sm) &= (rs)m &&(m\in M,\;r,s\in R)\\\end{aligned} symbol represents: associativity
0_M symbol represents: additive identity
-m symbol represents: inverse element
1_R symbol represents: multiplicative identity · characteristic of: base ring
\begin{aligned}r(m+n)&=rm+rn&&(m,n\in M,\;r\in R) \\(r+s)m &= rm+sm&&(m\in M,\;r,s\in R)\end{aligned} symbol represents: distributive property
Opposite of: comodule
Underlying structure: abelian group
Studied by: commutative algebra
ring theory
module theory
Has characteristic: base ring
length
radical of a module

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26 identifiers linking module across the web's reference databases

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statements

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Source last edited July 15, 2026. AIDB refreshes this page daily.

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AIDB. (2026). module (Q18848). Retrieved from https://aidb.si/e/module-Q18848