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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. And maybe it is worth observing that, if $R$ is an artinian ring (so, in particular, if $R$ is a skew field), then $T_n(R)$ is artinian (and hence noetherian); however, $T_n(R)$ is not von Neumann regular for $n \ge 2$ (or else the conclusion would have been more or less trivial).

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.)

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. And maybe it is worth observing that $T_n(R)$ is artinian (and hence noetherian), but is not von Neumann regular for $n \ge 2$ (or else the conclusion would have been more or less trivial).

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. And maybe it is worth observing that, if $R$ is an artinian ring (so, in particular, if $R$ is a skew field), then $T_n(R)$ is artinian (and hence noetherian); however, $T_n(R)$ is not von Neumann regular for $n \ge 2$ (or else the conclusion would have been more or less trivial).

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.

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

II could try to write to Schwiebert himself, but asking here might dig up some extra information.

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. And maybe it is worth observing that $T_n(R)$ is artinian (and hence noetherian), but is not von Neumann regular for $n \ge 2$ (or else the conclusion would have been more or less trivial).