# Abelian groups about rationals

Is the set $$\mathbb{Q}$$ under $$×$$ an abelian group? It is sure for $$\mathbb{Q} - {0}$$, but i think the whole set of rationals is not an abelian group as $$0 × a = a × 0 = 0$$, but the identity element is $$1$$ and i think it must be unique, and $$a$$ is not equal to $$0$$. Can you please help me whether it is an abelian group under $$×$$ or not?

It is not a group under multiplication, since $$0$$ has no inverse.