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maximal ideal

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A) Show that there is exactly one maximal ideal in Z_8 and in Z_9.
b) Show that Z_10 and Z_15 have more than one maximal ideal.

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We need to use the following fact.

I is a maximal ideal in a finite commutative ring R if and only if |R/I| is a prime number.
Because I is also a normal subgroup of R. If |R/I| is prime and I is not maximal, then we have some ideal J such that I is contained in J and J is contained in R. So I is a subgroup of J and J is a subgroup of ...

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