Ideal of a commutative ring with $30$ elements.

Question

Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of R and $|I|=10.$ Prove that $I$ is a maximal ideal.

Answer 1

If $J$ is an ideal between $I$ and $R$, with $|J|=n$, then we must have $10|n$ and $n|30$. Where does that leave us?

Answer 2

Although this problem can be solved applying the Lagrange theorem to the additive group of $R$, like paw88789 suggests, it looks like who wrote the exercise had Pedro Tamaroff's solution in mind. Indeed, while it is straightforward to see that $R/I$ must be a field, to be able to take this quotient you must know that $I$ is a two-sided ideal, which is immediate if $R$ is commutative.