---
tag: 7CHG6SD365
kind: concept
name: Subalgebra
msc: [16S99, 13B02, 08A30]
specializes: [676H7EGTKJ]
sources:
  - url: https://en.wikipedia.org/wiki/Subalgebra
same_as:
  wikidata: Q629933
  wikipedia: Subalgebra
---

## Summary

Let $A$ be an [[associative algebra|3CTW1P4HBP]] over a commutative ring $K$.
A subalgebra of $A$ is a subset that contains $1$ and is itself a $K$-algebra under the same operations.
It is a [[subring|0GNCRPYXYV]] of $A$ that is also a [[submodule|05WDSN597C]] for the scalar multiplication by $K$.
Subalgebras are the natural subobjects of algebras.
Images of algebra homomorphisms are subalgebras, and so are the [[subalgebras generated|2TQD8NMG9P]] by a few elements, such as $K[a] \subseteq A$.
It is the [[substructure|676H7EGTKJ]] for the operations of a $K$-algebra, with $1$ as a constant.

## Explanations

### L1 human

Take an algebra, a place where you can add, multiply and scale by numbers.
A subalgebra is a smaller piece that you never leave when you do these operations, and that contains $1$.
For example, in the algebra of all $2 \times 2$ real matrices, the upper triangular matrices form a subalgebra.
Adding, multiplying or scaling upper triangular matrices gives an upper triangular matrix, and the identity matrix is upper triangular.
The polynomials with integer coefficients do not form a subalgebra of the real polynomials $\R[x]$, because scaling $1$ by $1/2$ leaves the set.