A rather simple question. In my degree and in my own personal time I've been reading texts which use various algebras. Clifford, Grassman, Kac-Moody, Greiss, e.t.c. However I was wondering what is the formal definition of an algebra, i.e. what makes something an algebra. I know intuitively what the requirements are, but I'd like to hear the formal requirements, so as to understand what must exist in the first place for an algebra to be constructed.