Question

# Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I is a maximal ideal.

Commutative Algebra

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.

R is a commutative ring and $$|R|=30$$. If I is an ideal of R and $$|I|=10$$,
Then $$\displaystyle\frac{{R}}{{I}}$$ has order 3 and is thus isomorphic to $$\displaystyle\mathbb{Z}_{{3}}.$$
Since $$\displaystyle\mathbb{Z}_{{3}}$$ is field, I must be a maximal ideal.