Need the name of algebraic structure suitable for this

Question

If I want to prove that addition and multiplication are internal operations in the rational domain Q. What algebrac structure I need to prove that (Q,+, *) will be?

For example: (Q,+, *) need be a field, a ring ... or prove that (Q,+) and (Q, *) are groups both?

Answer 1

It's much less that any of that. All you need to prove is that addition and multiplication are binary operations from $\mathbb{Q}\times\mathbb{Q}$ into $\mathbb Q$ (that is, that $(\mathbb{Q},+)$ and $(\mathbb{Q},\times)$ are groupoids).