Decide whether the $\{a+b\sqrt{2} \mid a,b \in \Bbb Z\}$ is a ring with usual addition and multiplication. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field.

I am confused about how this is a ring closed under the binary operation of "+". If a = b = 1, then don't we have some number that is not in the integers? Thanks for your help!