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
ANSWERED
asked 2020-12-09

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.

Answers (1)

2020-12-10

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

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