Are Fields subsets of Integral Domains or is it the other way round?

Question

I recently came across a Theorem that mentioned "Every field is an integral domain." This makes me think that if F is any field and D is any integral domain, then $F \subseteq D$. But then I also encountered a line in an abstract algebra text that mentioned "every integral domain can be regarded as being contained in a certain field" which makes me think that $D \subseteq F$. How are these two propositions not contradictory? Are Fields subsets of Integral Domains or is it the other way round?

Answer 1

As you have interpreted them, the two statements are not contradictory: the two together imply $D=F$.

However, you have interpreted the first statement incorrectly.

"Every field is an integral domain" means that the set of fields is a subset of the set of integral domains, not that any particular field is a subset of any particular integral domain.

Your interpretation of the second statement is (mostly) correct though. The quibble is that the field $F$ depends on the integral domain $D$ in question, and is not arbitrary. This means: Integral domains are subsets of fields, not (necessarily) the other way around.

Answer 2

(There can be issues when you talk about sets which might be too large.)

But, every field is an integral domain. Not conversley, as for instance, $\Bbb Z$ (after which integral domains are named).

For the second statement, given an integral domain, we could, for instance, consider its field of fractions, which would contain it as a subdomain. There might also be other fields containing a given integral domain, and often are.