Term for an additive abelian group equipped with a multiplication which is distributive over the addition, but not necessarily associative?

Question

What is the official name of an additive abelian group with a biadditive multiplication (left and right distributivity of multiplication over addition and no other assumptions)?

Answer 1

I would call this a $\mathbb Z$-algebra.

EDIT: As pointed out in the comments, specifying that this algebra is non-unital and non-associative is incorrect. It is a not necessarily unital or associative algebra.

Answer 2

Apart from various versions of "definitions", I'd think the most effective term would exactly be "non-associative $\mathbb Z$-algebra".

That is, even though there are non-associative things (Lie algebras, Jordan algebras) that are literally called "algebras", outside of the context of "universal algebra", in the year 2020 I think most mathematicians would default into thinking associative, if you just say "algebra". That is, contrary to some notions of appropriate terminology, the modifier actually generalizes the base noun, rather than restricting it.

In my experience, most mathematicians in 2020 would hear "Lie algebra" not as a certain kind of "algebra" so much, as a monolithic identifier. Similarly with "Jordan algebra".

(If the context for your question really is about "universal algebra", that's a different matter, and I myself have no idea about current practice.)

In any event, to communicate effectively, it's surely best to tell your audience what you mean, and not depend on conventions (which change, in any case).

EDIT: and, to be clear, the general usage seems to be that saying something is a "non-X Y" means "not-necessarily-X Y". Yes, this leads to some silly situations, but it's typical. Also, although opinions would differ about such things, "universal algebra" does (in part) seek to understand "algebraic structures", and may have terminology of interest or use to you. And, yes, apparently there are strong feelings about terminology, so don't be surprised...

EDIT-EDIT: I feel somewhat duty-bound to make clear that in one form of this question, which is to ask what terminology would best communicate with most practicing mathematicians, it does not matter so much whether or not some mathematician uses a certain terminology, nor even whether it is somehow logically or grammatically correct. So the possibility of "justifying" terminology by finding some source that uses it is a mixed thing... because, again, if effective communication is the goal...