# Ring detail

## Name: Bergman's example showing that "compressible" is not Morita invariant

Description: Let $k$ be a field. Let $K$ be the field of rational functions for $k[t, x_{ij}\mid i,j=1,\ldots,4]$. Define $X_{pq}=(x_{p+i,q+j})_{i,j=1,2}$ for $p,q=0,2$. The ring $k[t]\langle X_{pq}\mid p,q=0,2 \rangle$ is a domain, and its ring of fractions $D$ is a division ring. Then $X=(X_{pq})\in M_4(K)$, and we can compute $X^{-1}\begin{bmatrix}I&0\\0&0\end{bmatrix}X=\begin{bmatrix}E_{00}&E_{02}\\E_{20}&E_{22}\end{bmatrix}$ where $E_{pq}\in M_2(D)$. The ring $R$ is $k\langle E_{pq}, tE_{pq}\mid p,q=0,2\rangle\subseteq D$. ($M_2(R)$ is not compressible, although $R$ is.)

Reference(s):

• G. M. Bergman. Some examples of non-compressible rings. (1984) @ Section 2 pp 5-7

Legend
• = has the property
• = does not have the property
• = information not in database
Name Measure
composition length left: $\infty$right: $\infty$
Name Description
Idempotents $\{0,1\}$
Nilpotents $\{0\}$
Zero divisors $\{0\}$