I'm working with elementary rings, and my professor gave me about ten of these to start but it seems like a lot of work with how he managed it. I know you guys don't answer homework so I chose one my professor solved to help me compare so I can do the others. Any help would be greatly appreciated, the groups were easy but the rings are a little more difficult for me to grasp.

Consider the set R = {a + b (SQRT 2): a, b are in Z}

Prove that R is a subring ofR.

Is R commutative?

Does R have identity?

Is R an integral domain?

Is R a division ring?

Is R a skew field?

Is R a field?