What is an Algebra? Formal Requirements Explained

[Son Goku]

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.

 

[matt grime]

An algebra is a vector space over some field, k, with a multiplication that behaves reasonably. This is sort of wooly, I admit. Normally (i.e. 99.9% of the time) we mean that it is a Ring so the multiplication operation is associative. Some times these days people drop the requirement of associativity and speak of associative algebras to mean a vector space that is simultaneously a ring.

 

[Son Goku]

An algebra is a vector space over some field, k, with a multiplication that behaves reasonably. This is sort of wooly, I admit. Normally (i.e. 99.9% of the time) we mean that it is a Ring so the multiplication operation is associative. Some times these days people drop the requirement of associativity and speak of associative algebras to mean a vector space that is simultaneously a ring.

Thank you.
That basically answers my question.