Is $M(2,\mathbb Z)$ a Von Neumann regular ring?

Question

Is it true that $M(2,\mathbb Z)$ is a Von Neumann regular ring ? https://en.wikipedia.org/wiki/Von_Neumann_regular_ring

Answer 1

You mean $M_2(\mathbb Z)$?

No. Von Neumann regularity is Morita invariant, and $\mathbb Z$ is not von Neumann regular.

Another, more elementary way to see it: You can't find a matrix satisfying this equation. $$ \begin{bmatrix}2&0\\0&2\end{bmatrix}\begin{bmatrix}w&x\\y&z\end{bmatrix}\begin{bmatrix}2&0\\0&2\end{bmatrix} \\=\begin{bmatrix}4&0\\0&4\end{bmatrix}\begin{bmatrix}w&x\\y&z\end{bmatrix} \\=\begin{bmatrix}2&0\\0&2\end{bmatrix} $$