# Ring detail

## Name: Shepherdson's domain that is not stably finite

Description: Let $k$ be a countably infinite field and $X=\{s,t,u,v,w,x,y,z\}$ be indeterminates. Let $R$ be the free algebra over $k$ 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

Reference(s):

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

Legend
• = 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 $\{0,1\}$
Nilpotents $\{0\}$
Zero divisors $\{0\}$