About the ring $\{a+b\sqrt{2}\mid a,b\in \Bbb Z\}$ 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!

The condition is that the coefficients are integers, not that the elements of the ring are integers. The element $1+\sqrt 2$, in your example, is perfectly valid because the coefficients, $a=1$ and $b=1$, are integers. On the other hand, something like
$$
\frac{1}{2}-\frac{1}{2}\sqrt2
$$
is not in the ring because it has coefficients that are not integers.
