[Math] Does every element of an integral domain have an inverse

Question

I am reading a first course in algebra 7th edition written by John B. Fraleigh. I have seen the following two definitions:

1) A field is a commutative ring in which every nonzero element has
multiplicative inverse.

2) An integral domain is a commutative ring with unity 1 and
containing no zero divisors.

Then i saw a picture in the book that shows fields as subsets of integral domains like in the following picture:

My question is, how do we understand from these two definitions that fields are subsets of integral domains? In the definition of integral domain, i do not see anything saying that every element in the ring should have an inverse, it just says that there must be a multiplicative identity. Am i missing something or is there something missing in the definitions?

Thank you

Answer 1

No, it is not necessarily the case that every element in an integral domain has a multiplicative inverse.

Every field is an integral domain, but not every integral domain is a field. Hence we have that the set of all fields is a proper subset of the set of all integral domains.