Subalgebra

Summary

Let $A$ be an associative algebra 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 of $A$ that is also a submodule 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 by a few elements, such as $K[a] \subseteq A$. It is the substructure for the operations of a $K$-algebra, with $1$ as a constant.