Can $\mathbb{Z}_{6}$ be a subring of some field?

Question

I think the answer is yes because $\,\mathbb Z_{6}\,$ is a ring, it has a unity and has multiplicative inverse and its elements are commutative so it can be a subring of a field. Is this correct? What I did was that I moved through the definitions ring w/unit and progressed to that of the field via division ring.

Answer 1

Your assertion that every element of $\mathbb{Z}_{6}$ has an inverse is not correct. To see this, note that $2\cdot 3 = 0$ in $\mathbb{Z}_{6}$. Can zero divisors have inverses? Can you have zero divisors in a field?

Note that there is something more general going on here structurally. The characteristic of a ring $R$ is defined to be the generator of kernel of the unique ring homomorphism $\phi: \mathbb{Z} \rightarrow R$. It is a simple exercise to prove that for any subring $S$ of $R$, $S$ must have the same characteristic as $R$. It is another basic exercise to prove that the characteristic of any field (in fact, domain) is either $0$ or prime. This amounts to understanding the hints above; if a ring $R$ has composite characteristic, then it necessarily has zero divisors.

Together, these show that the characteristic of any subring of a field is either $0$ or prime (really, you only need the second fact, since every subring of a field is a domain). In the specific example above, the characteristic of $\mathbb{Z}_{6}$ is $6$, which is composite, and thus $\mathbb{Z}_{6}$ cannot be a subring of any field.

Answer 2

A ring is a subring of a field iff it is a domain.

Your ring is not a domain: $2\cdot3=0$.