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:

enter image description here

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

The answer to your title question is no. This (not amazingly useful) picture means that every field is an integral domain. So assume $F$ is a field and show it contains no zero divisor. — Julien, May 12 '13 at 14:45
@CutieKrait , do you mean every field is an integral domain? — Dan Rust, May 12 '13 at 14:54
every finite integral domain is a field and thus has inverse for each element — Bhaskar Vashishth, Feb 06 '16 at 04:46

score 8 · Accepted Answer · answered May 12 '13 at 14:52

8

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.

answered May 12 '13 at 14:52

amWhy

209,954

Thank you, i understand now – Yasin Razlık May 12 '13 at 14:53
@amWhy: that is excellent feedback! :-0 +1 – Amzoti May 13 '13 at 00:30
Yes, that's always rewarding, @Amzoti! – amWhy May 13 '13 at 00:31

score 2 · Answer 2 · answered May 12 '13 at 15:44

As already noticed, every field is an integral domain but the converse doesn't hold (take for example $\mathbb{Z}$). In order to show that a field $F$ is an integral domain, suppose $a,b\in F$ are such that $ab=0$ and assume $a$ is not zero. Then since every non zero element in a field is invertible, one has $$ab=0\implies a^{–1}ab=0\implies b=0. $$ Similarly if $b\neq 0$ one gets that a must be zero and hence $F$ is an integral domain.

Does every element of an integral domain have an inverse?

2 Answers2