##  [Module](/module) 

  ##  [Module](https://natural.quantumdictionary.io/module-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebraic structure consisting of an abelian group (M, +) together with an action of a ring R (not necessarily commutative or a field) on M that is distributive and associative with respect to ring multiplication and respects the ring identity if present; modules generalize vector spaces by allowing scalars from rings.

 

 

 

 

 





 

 



 ##  [Module](https://mathlogic.quantumdictionary.io/module-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A structure M equipped with an abelian group operation (addition) and an action of a ring R (scalar multiplication) that is compatible with ring multiplication and addition: r·(m+n) = r·m + r·n, (r+s)·m = r·m + s·m, and (rs)·m = r·(s·m).

 

 

 

 

 





 

 



 ##  [Module](https://puremath.quantumdictionary.io/module-2) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An R-module is an abelian group equipped with a compatible action of a ring R: a map R × M → M satisfying distributivity, associativity with ring multiplication, and that 1·m = m when R has unity. It generalizes vector spaces by allowing scalars from a ring.

 

 

 

 

 





 

 



 ##  [Module](https://algebra.quantumdictionary.io/module-3) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An algebraic structure consisting of an abelian group (the underlying additive group) together with an action of a ring R (with unity) on that group satisfying distributivity, associativity with ring multiplication, and identity action; a module is a direct generalization of a vector space where scalars come from a ring instead of a field.