# What is the difference between a Ring and an Algebra?

In mathematics, I want to know what is indeed the difference between a ring and an algebra?

A ring $R$ has operations $+$ and $\times$ satisfying certain axioms which I won’t repeat here. An (associative) algebra $A$ similarly has operations $+$ and $\times$ satisfying the same axioms (it doesn’t need a multiplicative identity, but this axiom isn’t always assumed in rings either), plus an additional operation $\cdot\;\colon R\times A\to A$, where $R$ is some ring (often a field) that satisfies some axioms making it compatible with the multiplication and addition in $A$. You should think of this as an analogue of scalar multiplication in vector spaces.