Ring $R_{ 169 }$

Description:

Let $K$ be a commutative integral domain (we take $K = \Bbb Q$). Define two countably sets of variables $A = \{a_i, b_i, c_i, d_i\}$ and $X = \{x_i, y_i, u\}$, where $i$ runs through positive integers. Consider the graded $K$-algebras $T = K[A]$, $D = K[X]$ where the indexed variables have degree $1$ and $u$ has degree $0$. The kernel $P$ of the homomorphism $f$: $T \to D$, $a_i \mapsto ux_i$, $b_i \mapsto x_i$, $c_i \mapsto y_i$, $d_i \mapsto uy_i$, is a homogeneous prime ideal of $T$; let $P_i$ be its homogeneous component of degree $i \ge 0$. The component $P_2$ is a free $K$-module with basis $B = \{\gamma_{ij} = a_i c_j - b_i d_j, a_i b_j - a_j b_i, c_i d_j - c_j d_i\}$ for all $i,j$. Define the ideal $I \lhd T$ generated by the union of the sets $\{ P_i\mid i\ge 3\}$, $B \setminus \{ \gamma_{ii}\mid i\ge 1 \}$, and $\{\gamma_{ii} - \gamma_{jj}\mid \text{ for all } i,j\}$ . The ring is $R = T/I$.

Notes: $M_2(R)$ does not satisfy ACC annihilator on either side.

Keywords polynomial ring quotient ring