Camillo and Nielsen's McCoy ring

Construction:

Let $K = \Bbb F_2$. Take four disjoint sets of variables $\Sigma_a$, $\Sigma_b, \Sigma_c, \Sigma_d$, each indexed by $\Bbb N \cup \{ 0 \}$, and denote their union by $\Sigma_0$. Let $A_0 = K\langle \Sigma_0 \rangle$ be the free $K$-algebra over these variables. Set $f(x) = \sum\limits_{i = 0}^\infty (a_i + b_i x) t^i$, $g(x) = \sum\limits_{i = 0}^\infty (c_i + d_i x) t^i \in A_0[[t]][x]$. Take $C_0$ to be the set of $A_0$-coefficients of each monomial in $f(x) g(x)$, and set $B_0 = A_0/(C_0)$ (the quotient by the ideal generated by $C_0$). Define inductively:

• $\Sigma_{i+1} = \Sigma_i \sqcup \{ x_S, y_S: S \subset \Sigma_i, |S| < \infty\}$ (for every *finite subset* of old variables, adjoin a *pair* of new variables);
• $A_{i + 1} = K\langle \Sigma_{i + 1} \rangle$;
• $C_{i + 1} = C_i \cup \{ x_S a_i, x_S b_i, c_i y_S, d_i y_S, x_S s, s y_S: S \in \Sigma_{i + 1} \setminus \Sigma_i, s \in S \}$,
• and $B_{i + 1} = A_{i + 1}/(C_{i + 1})$.

The ring $R = R_{163}$ is the direct limit $\bigcup\limits_{i=0}^\infty B_i$.