Is $R$ an integral domain? Let $R = \{a + b\alpha |\ a,b \in \mathbb{Z}\}\subseteq \mathbb{C}$ where $\alpha = \frac{1}{2}(1+\sqrt{-19})$
Is $R$ an integral domain?
To show whether or not $R$ is an integral domain, letting $x = \{a+b\alpha\}$ and $y = \{c+d\alpha\}$, where $a,b,c,d \in \mathbb{Z}$ and showing $xy = yx \in R$ would suffice without going into further, since an integral domain is a commutative ring at first?

Hint $\ $ One easily verifies that it is a subring of $\,\Bbb C\,$ since it satisfies the subring test, i.e. it contains $\,1\,$ and is closed under subtraction and multiplication (use $\,\alpha^2 = m\alpha + n\,$ for some $\,m,n\in\Bbb Z).\,$ Therefore, being a subring of a field, it is an integral domain.
