# 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.

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.

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R is a commutative ring and $$|R|=30$$. If I is an ideal of R and $$|I|=10$$,
Let R be an order 30 commutatie ring, and I be an ideal of R with order 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.
Hence, proved