infinite product of finite Noetherian rings

Question

i know that if two rings are noetherian then their product is noetherian. Can I apply this to the infinite case say: $ℤ_2 x ℤ_3 x ℤ_4 x ...$ and say that this is noetherian since each $ℤ_n$ is finite and therefore noetherian?

Answer 1

For each $n$ the following

$$I_n = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}\times ...\times \mathbb{Z}/n\mathbb{Z} \times \{0\}\times ...\times \{0\} \times ...$$

is an ideal. Thus $\{I_n\}_{n\geq 1}$ is an infinite and increasing sequence of ideals. Hence your ring is not noetherian.