Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of $R$ and $|I|=10$, prove that $I$ is maximal ideal
Solution: $|R/I|=3 \implies R/I \approx Z_3$ which is a field.
If $R$ is a commutative ring with unity and $I$ is an ideal, then $R/I$ is a field if and only if $I$ is a maximal ideal. Hence, in this problem, $I$ is a maximal ideal iff $R$ contains a unity.
How do we prove that $R$ must contain a unity?
I know that a finite commutative ring with no zero divisors definitely contains a unity. But, then $R$ has not been stated to not contain zero divisors either.
Thank you for your help..