[Math] Example of an integral domain which is not a field

abstract-algebraintegral-domainring-isomorphismring-theory

I just proved that a commutative ring $R$ is an integral domain iff $R$ is isomorphic to a subring of a field.

My question is why can't $R$ be a field with these conditions? Aren't we satisfying all of the field's properties?

Thanks.

Best Answer

For a counter-example, let's have a look at $\mathbb{Z} \subseteq \mathbb{Q}$.
Here $\mathbb{Z}$ is an integral domain which is not a field;
also you can check that $\mathbb{Z}$ is a sub-ring of the field of rational numbers $\mathbb{Q}$.

Note that $\mathbb{Z}$ satisfies all of the field's properties; except the property which concerns the
existence of multiplicative inverses for non-zero elements. For example $2^{-1} \notin \mathbb{Z}$ .

Related Question