I am trying to find nonzero prime ideals of [itex]\mathbb{Z} \oplus \mathbb {Z}[/itex], specifically those which are not also maximal.(adsbygoogle = window.adsbygoogle || []).push({});

If I try to do direct sums of prime ideals, the resulting set is not a prime ideal. (e.g., [itex]2 \mathbb{Z} \oplus 3 \mathbb{Z}[/itex] is not prime since [itex](3,3) \cdot (2,2) = (6,6)\in 2 \mathbb{Z} \oplus 3 \mathbb{Z}[/itex], but [itex] (2,2),(3,3) \notin 2 \mathbb{Z} \oplus 3\mathbb{Z} [/itex].)

In fact, I don't think a prime ideal could be constructed in this way since I can always take a product of the form (1,x)(y,1) and obtain (y,x) and since 1 isn't a multiple of any integer other than 1, neither of the factors would have come from the ideal.

Can somebody please help me find the prime ideals? Thanks.

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# Prime Ideals of direct sum of Z and Z

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