# Ring detail

## Name: Dedekind finite, not stably finite ring

Description: Let $k$ be a 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 $B=\begin{bmatrix}w&x\\ y&z\end{bmatrix}$.

Krull dimension: (unknown)

Keywords quotient ring

Reference(s):

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