Hi all, I'm trying to understand the structure of an "algebra over a field".

The definition I get from wolfram is

"Formally, an algebra is a vector space V over a field F with a multiplication."

So if I was to break down this down....

A vector space has two operations

+:V x V --> V (addition)

. :F x V --> V

and so if A was an algebra over F, would it mean it has three operations?

+:A x A --> A (addition)

. :A x V --> A

* : A x A --> A (multiplication)