Ring $R_{ 35 }$

Shepherdson's domain that is not stably finite


Let $X=\{s,t,u,v,w,x,y,z\}$ be indeterminates. Let $R$ be the free algebra over $\mathbb Q$ and $X$ modulo relations which make the matrix equation $AB=I_2$ hold, where $A=\begin{bmatrix}s&t\\ u&v\end{bmatrix}$ and $B=\begin{bmatrix}w&x\\ y&z\end{bmatrix}$.

Keywords free algebra quotient ring


  • T.-Y. Lam. Lectures on modules and rings. (2012) @ Ex 18, p 19

  • = has the property
  • = does not have the property
  • = information not in database
Name Measure
cardinality $\aleph_0$
composition length left: $\infty$right: $\infty$
Name Description
Idempotents $\{0,1\}$
Left singular ideal $\{0\}$
Left socle $\{0\}$
Nilpotents $\{0\}$
Right singular ideal $\{0\}$
Right socle $\{0\}$
Zero divisors $\{0\}$