1. Maximal Ideal 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.

Im pretty sure I understand why its maximal, its because if there was an ideal that properly contained I then the order would be greater than 10 but the order of all proper sub-rings of R would be positive divisors which stop at 10. ok my problem is proving this elegantly on paper... any thoughts. thanks

2. Well, there's also 15, but you can rule that out, because...?

3. Originally Posted by fizzle45
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.

Im pretty sure I understand why its maximal, its because if there was an ideal that properly contained I then the order would be greater than 10 but the order of all proper sub-rings of R would be positive divisors which stop at 10. ok my problem is proving this elegantly on paper... any thoughts. thanks

More eleganty than the above? You can remark that R is an (additive) group and an ideal is a subgroup of it so Lagrange's Theorem applies...which ammounts to what you said.

Tonio

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R be a commutative ring with order 30 if I is ideal of T with order 10 prove that I is maximal

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