Localization of integral domains is monotone

Question

Assume that $A \subseteq B$ are two integral domains and let $\mathbb{K}_A$ and $\mathbb{K}_B$ their corresponding fields of fractions. Seeing some easy examples, it seems that $\mathbb{K}_A \subseteq \mathbb{K}_B$. However, we work with equivalence classes, so it is not so trivial to get inclusion. Is there an example for which the previous inclusion fails to hold? Is there an embedding in this case?

Answer 1

The field of fraction $\mathbb K_A$ has the universal property that any ring morphism from $A$ to field $K$ uniquely extends to a morphism $\mathbb K_A\to K$. The inclusion $A\hookrightarrow B$ gives an inclusion $A\hookrightarrow \mathbb K_B$ and a field morphism $\mathbb K_A\to\mathbb K_B$, which is automatically injective.