Prime and Maximal Ideals of $\mathbb{Z}_2 \times \mathbb{Z}_4$

Question

I'm having some trouble finding ideals in general. The problem I'm stuck on is:

Find all prime and maximal ideals of $\mathbb{Z}_2 \times \mathbb{Z}_4$.

I know that a finite integral domain is a field, which means the prime and maximal ideals are the same. That means I just have to find one or the other, right? But I don't really know how to go about finding ideals for $\mathbb{Z}_n \times \mathbb{Z}_m$. Any help would be great. Thank you!

http://math.stackexchange.com/questions/101214/structure-of-ideals-in-the-product-of-two-rings This should definitely help you get started. There aren't very many ideals to consider. — LASV, Nov 21 '16 at 00:46

T C · Accepted Answer · 2018-05-17T06:53:56.950

-2

An ideal of $Z_2 \times Z_4$ must have the form $I \times J$ where $I$ is an ideal of $Z_2$ and $J$ is an ideal of $Z_4$, both of which are not necessarily proper. So clearly there's not so much ideals to try!

edited May 17 '18 at 06:53

answered Nov 21 '16 at 15:30

T C

2,563

You should clarify that the ideals $I$, $J$ need not be proper -- one can be the whole ring. – BallBoy May 04 '18 at 00:47

Prime and Maximal Ideals of $\mathbb{Z}_2 \times \mathbb{Z}_4$

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