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?