Closed dyunov closed 3 months ago
Regarding compressibility of $T_n(k)$. The idempotents are all conjugate (in $T_n$, not $M_n$!) to diagonal ones: use the "elementary similarity transformations" like this
to zero out every element $a{ij}$ s.t. $a{ii} = 1$, $a_{jj} = 0$ or vice versa (and do it row by row, from left to right in every row). Then the result has to be diagonal, otherwise it would not be diagonalizable at all (even in $M_n$) and therefore not idempotent.
For the diagonal idempotents the compressibility claim is clear, as these $eRe$ are also just triangular rings, the scalar matrices being their center.
I plan on adding this in conjunction with another direct limit of matrix rings based on the whole partial order of divisibility (instead of just the chain selected here.) I'm waiting to chat with my advisor who I believe used such a ring in his dissertation.
Description
This ring is Ex1.2 from "Structure and topological conditions of NI rings" by Hwang, Jeon, Lee (2006). In Ex1.3 they use an arbitrary domain instead of $\Bbb Q$ (and in some papers that use this construction they need characteristic 2, but not here). This is clearly a subring of $RCFM(\Bbb Q)$ (as $2^k$-periodic block diagonal matrices) and, in turn, of $R_{15}$; but also of the ring in suggestion #65.
The units are again the elements with nonzero numbers on the diagonal (because it is so in $A_n$'s), the Jacobson radical is everything above diagonal. It is nil but not nilpotent.
Properties