The ring of upper triangular $n$-by-$n$ matrices over a skew field is (left and right) Rickart

Question

Let $T_n(R)$ be the ring of upper triangular $n$-by-$n$ matrices with entries in a (commutative or non-commutative) unital ring $R$. It happened to me to note that, if $R$ is a skew field, then $T_2(R)$ is right Rickart, i.e., the right annihilator of every element is a (right) direct summand (and I think I can generalize this to $T_n(R)$). So I started looking around and found on Ryan C. Schwiebert's Database of Ring Theory (here) that $T_n(R)$ is in fact known to be a (left and right) Rickart ring provided $R$ is a field. Thence, the question is:

Do you have a reference for this last result (or for a more general one covering the case where $R$ is a skew field)?*

I could try to write to Schwiebert himself, but asking here might dig up some extra information. (Maybe it is worth observing that $T_n(R)$ is not von Neumann regular for $n \ge 2$, or else the conclusion would have been more or less trivial.)

Answer 1

By https://encyclopediaofmath.org/wiki/Rickart_ring , a left Rickart ring is characterised by all principal left ideals being projective. But your ring (in the skew-field case as in the title of your question) is a hereditary ring so every submodule of a projective module is also projective. Thus every principal left ideal is projective and your ring is Rickart.

That your ring is hereditary (meaning global dimension at most one) is easy to see by looking at the projective dimensions of the simple modules and using Auslander's result that the global dimension is equal to the supremum of the projective dimensions of the simples.