I have only been able to determine three ideals: A maximal that is $\Bbb Z\times 2\Bbb Z$. A prime $\Bbb Z\times \{0\}$ Finally a proper ideal $\Bbb Z\times 4\Bbb Z$. I don't know if there will be more ideals, some help to be able to determine them all.
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This may help: https://math.stackexchange.com/questions/3345579/proof-about-ideals-of-cartesian-product-of-rings. Similarly: https://math.stackexchange.com/questions/101214/structure-of-ideals-in-the-product-of-two-rings?rq=1 and https://math.stackexchange.com/questions/3583208/every-ideal-of-the-ring-r-times-s-where-r-and-s-are-rings-is-the-cartesian?noredirect=1&lq=1 – morrowmh May 10 '22 at 02:40
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Also asked yesterday (now deleted by the author). If it's a classmate maybe you two should start a study group. – rschwieb May 10 '22 at 13:18