These "multiple occasions" may be implicitly applying the subring test, i.e. $\,\Bbb Z[\sqrt 2] \subseteq \Bbb R\,$ contains $\,1_{\Bbb R}$ and is easily verified to be closed under subtraction and multiplication hence it is a subring of $\,\Bbb R.\,$ Furthermore, it is clear that nontrivial subrings of domains remain domains (since the inference$\,a,b\neq0\,\Rightarrow\, ab\neq 0 $ necessarily remains true in every subring containing $\,a,b)$
Remark $ $ In fact, by the general definition of ring adjunction, $\,\Bbb Z[\sqrt 2]\subseteq\Bbb R$ is the intersection of all subrings of $\Bbb R$ containing $\,\Bbb Z\,$ and $\,\sqrt 2\,$ so it is a domain, being a nontrivial intersection of domains.
Such inferences are quite common in algebra, because we define (minimally) "generated" structures via such intersections, and the (universal) logical form of the axioms of many algebraic structures makes it immediately clear that they are closed under intersections and subalgebras (e.g. the $\rm\color{#c00}{universal}$ form ring axiom $\,\color{#c00}{\forall x,y:}\ x+y = y+x).\,$ This is made more precise when studies the relationship between syntax and semantics in universal algebra and model theory.