Well, to verify that $R$ is a subring of $\Bbb C$, you'll need to verify that:
- $R$ is a subgroup of $\Bbb C$.
- $R$ is closed under multiplication.
- $R$ contains the multiplicative identity. (This axiom is to be verified if you consider the multiplicative identity as a part of the structure of your ring.)
To verify the first property:
- Note that $R$ is non-empty as it contains $\Bbb Z$ inside it, when you set $y=0$ in your definition.
- Let $x_1+\alpha y_1$ and $x_2 + \alpha y_2$ be elements of $R$. Clearly, $x_1-x_2+\alpha(y_1-y_2) \in R$.
To verify the second property:
- $(x_1+\alpha y_1)(x_2+\alpha y_2)=x_1x_2+\alpha(x_1y_2+y_1x_2)+\alpha^2y_1y_2$. Now, can we write $\alpha^2$ in terms of $\alpha$ and other integers? If we could it would follow that $R$ is closed under multiplication.
A brief thought and acquaintance with complex numbers tell us that $-\alpha$ is a complex cube root of unity. And, it is know that, the cube roots of unity satisfy $1 +(- \alpha) +(-\alpha)^2=0$. This means, $$ \alpha^2=\alpha-1$$ So the product is, $$(x_1+\alpha y_1)(x_2+\alpha y_2)=x_1x_2-y_1y_2+\alpha(x_1y_2+y_1x_2+y_1y_2) \in R$$
To verify property $3$:
- Note that by setting $x=1$ and $y=0$, you have $1 \cdot (x_1+\alpha y_1)= x_1 +\alpha y_1$. So, $1$ is the neutral element in $R$.