Let $\varphi: R \rightarrow S$ be a NOT surjective ring homomorphism. Let $I$ be an ideal of $R$. Then it can be easily proved that $\varphi(I)$ is an ideal of $\varphi(R)$. I know a counterexample when $\varphi(I)$ is not an ideal of $S$ (when $R = \mathbb{Z}$ and $S = \mathbb{Z}[x]$). But can it happen that $\varphi(I)$ will be an ideal of $S$ for a NOT surjective $\varphi$? Could you show me an example, please?
2026-03-25 01:17:25.1774401445
The image of an ideal by a ring homomorphism
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