I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field.
But until now, I haven't asked myself about infinite upper triangular matrix rings over a field (sides indexed by some infinite set $I$ which has a linear order.) "Upper triangular" means, of course, that the $i,j$ entries are zero if $i >j$. Let's call the ring $R$, and assume $I=\Bbb N$ for simplicity.
This ring can be realized as a subring of the linear transformations of a countable dimensional vector space in the following way: select a chain of subspaces $V_i$ of dimension $i$ for each natural number $i$, with $V_i\subseteq V_{i+1}$ for all $i$. The set of transformations which map $V_i$ into $V_i$ for all $i$ is represented by the ring we are speaking of.
What can be said about $R$'s ring-theoretic properties? It seems to satisfy quite few positively.
Structurally:
It is clearly a subring of the ring of column-finite matrices indexed with $I$.
it seems the Jacobson radical $J(R)$ is still the strictly upper triangular matrices. The Jacobson radical isn't nilpotent anymore when $I$ is infinite (it is not even nil).
- I have read that the elements with no zeros on the diagonal are invertible. (Perhaps these are all the units?)
- It looks like the powers of $J(R)$ form an infinitely long filtration of ideals with intersection $\{0\}$.
Some observations on properties:
- It's certainly not Artinian or Noetherian on either side.
- It isn't even semiprime: $e_{12}Re_{12}=0$ for the matrix units $e_{ij}$.
- Unlike the ring of column-finite matrices, the ring of upper triangular matrices is Dedekind finite since it has a commutative quotient ring (namely $R/J(R)$.)
To give a few concrete questions:
- is it hereditary, semihereditary or coherent on either side?
- is it stably finite?
- What do its socles look like?
- What do its left/right singular ideals look like?
I can split the question into two halves, perhaps, if the need arises.
