Have you been given the definition of a ring? If not then there is a list of properties that must hold ont he wiki page: https://en.wikipedia.org/wiki/Ring_%28mathematics%29#Definition_and_illustration/wiki/Ring_%28mathematics%29
For example, is there an additive identity.
Well, 0 would seem like an obvious choice.
Is it true that $0 +(a+b\sqrt 2 ) = a+b\sqrt 2$?
It is, since $0 +(a+b\sqrt 2 ) = (0+a) + (0+b\sqrt 2) = a+b\sqrt 2$.
You need to check all the parts of the definition in similar manner.
Feel free to ask in the comments if any parts of the definition don't make sense.
Edit: As pointed out in the comments, you do not need to prove that 0 is the identity since you are proving it is a subring, rather than just proving it is a ring. It should still serve as an example for proving the other parts of the definition are satisfied though.