##  [Direct Sum](/direct-sum) 

  ##  [Direct Sum](https://puremath.quantumdictionary.io/direct-sum-0) 

  

 [![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

A construction that forms a new object from a family of objects by taking tuples and allowing only finitely many nonzero components (in the usual algebraic finite direct-sum case), with componentwise operations and injections from each summand.

 

 

 

 

 





 

 



 ##  [Direct Sum](https://algebra.quantumdictionary.io/direct-sum-1) 

  

 [![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

A construction that combines a family of modules or groups into a larger object consisting of tuples with finitely many nonzero components, with operations defined componentwise; it serves as the coproduct in the category of modules.