# Difference between a ring with 1 and an associative algebra

I'm reading Clausen, Fast Fourier Transforms, which deals with FFTs mostly in groups, abelian and non-abelian, and especially the symmetric group.

He defines an algebra as a complex vector space with a binary multiplication such that it is a ring with 1.

What confuses me is that I don't really understand the difference between this algebra (which is an acciative algebra, I guess) and a ring with 1. Yes, his algebra is defined to be a vector space, but is that it? I'm asking, since I never read something like "an algebra is a ring that's also a vector space".

Does that make any sense?

-

Usually you would define a $k$-algebra (for $k$ a field) like this:
An algebra $A$ is a ring that is also a vector space such that $\lambda (ab)=(\lambda a)b=a(\lambda b)$ for $\lambda\in k$ and $a,b\in A$, i.e. scalar multiplication and ring multiplication should be compatible.