O'Meara's infinite matrix algebra
Recently (A new setting for constructing von Neumann regular rings by K. C. O'Meara, Communications in Algebra 45.5 (2017): 2186-2194) O'Meara described an algebra that is very useful for understanding Bergman's unit regular ring, and is very interesting in its own right.
Let $F$ be a field, and define the following $F$ vector spaces: $$ M = \text{ the set of all $\omega\times \omega$ matrices over } F \\ U = \text{ the ring of row-finite $\omega\times\omega$ matrices over } F \\ V = \text{ the ring of column-finite $\omega\times\omega$ matrices over } F \\ N = \text{ the set of all $\omega\times\omega$ matrices with finitely many nonzero entries } $$ O'Meara's algebra is $B=\begin{bmatrix}U&M\\N&V\end{bmatrix}$.
It is noted (by O'Meara, who attributes it to Bergman) this can be viewed as a ring of functions. Let $X$ be the space of countably infinite column vectors over $F$, and $Y$ be the subspace of $X$ whose elements only have finitely many nonzero entries. Give $X$ the product topology and $Y$ the discrete topology, and let $Z=X\times Y$ as a topological space. Then $B$ can be identified with the algebra of continuous linear transformations of $Z$.
Three features are useful for connecting this with Bergman's unit-regular ring.
- The socle $soc(B)=\begin{bmatrix}soc(U)&soc(U)M+Msoc(V)\\ N&soc(V)\end{bmatrix}$.
- There is a ring homomorphism $\pi:B/soc(B)\to \begin{bmatrix}U&0\\0&V\end{bmatrix}$ projecting onto the diagonal.
- There is a natural copy of the Laurent series $k((t))$ in $B$:
$ b_mt^{-m}+\ldots+b_1t^{-1}+\sum_{i=0}^\infty c_it^i$ maps to
The image of the Laurent series in $B$ will be called $K$.
O'Meara shows that $\pi(K)+soc(B)$ is isomorphic to Bergman's unit-regular ring $R_{81}$, and the subring $\pi(K)\oplus \begin{bmatrix}soc(U)&0\\0&soc(V)\end{bmatrix}$ is isomorphic to Bergman's non-unit-regular subring $R_{82}$.