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?