Ring $R_{ 127 }$

Description:

Let $k$ be a countable field of characteristic other than $2$. Let $w=\sum a_ix^i\in k[[x]]$ which is transcendental over $k(x)$. Let $x,y,z$ be algebraically independent over $k$, and set $z_1=z$, and $z_{i+1}=[z-(\sum_{j\lt i}a_jx^j)^2]/x^i$. Let $S$ be the ring $k[x,y,z_1,z_2,\ldots]$ localized at the ideal generated by $x,y$ and the $z_i$'s. The required ring is $S[W]/(W^2-z)$.

Keywords localization polynomial ring power series ring quotient ring