What is an example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$? I've tried constructing one, but all my attempts have failed...
2026-05-04 21:13:39.1777929219
example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$
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Let $k$ be a field and consider the polynomial ring $R=k[x]$. $R$ is an integral domain and so $Ass_R(R)=\left\{0\right\}$. Moreover, the ideal $(x)$ is maximal. Now take $I = (x^2)$. The $R$-module $R/I = k[x] / (x^2)$ is annihilated by a power of $(x)$. In fact, $(x)^2 \subset Ann_R(R/I)$. This shows that $Ann_R(R/I)$ is $(x)$-primary. We conclude that $Ass_R(R/I)=\left\{(x)\right\}$.