Ring $R_{ 208 }$

Description:

For each $n >1$, define $R_n=\mathbb Q[x_n,y_1,\ldots y_n]/(\{x_ny_i\mid 1\leq i\leq n\})$. $R_i$ is Noetherian, hence coherent. Furthermore, the annihilator of $x_n$ is generated by no fewer than $n$ elements. Let $R$ be the ring $\prod_{i=1}^\infty R_i$. (The annihilator of $(x_1,x_2,x_3,\ldots)$ is not a finitely generated ideal.)

Keywords direct product polynomial ring quotient ring