Let $[F':F]=n$ be a field extension, $S \supset S'$ the full $F$- and $F'$-linear rings of the countable-dimensional $F'$-vector space $V$, and their ideals $I \supset I'$ of transformations with finite-dimensional range. Then the sum $T = S' + I$ is a subring in $S$.
The idempotent $e = E_{11}$ of $S'$ is primitive in $S'$ and is a sum of $n$ primitive orthogonal idempotents $e_1, \ldots, e_n$ in $S$ and $T$. Let $C$ be the corner ring $(e_n + 1 - e) T (en + 1 - e)$. Then the ring $R{83}$ is defined as the set of all countable-sized column-finite matrices $M$ over $C$ with the following property: there exists an integer $N = N(M)$ such that the row $M{N,*} = [m{N1}, m{N2},\ldots]$ has all $m{Ni} \in F = F1 \subset C$, and all the next rows are shifts of $M{N,*}$, namely, $M{N + k, *} = [0, \ldots, 0, m{N1}, m{N2},\ldots]$ ($k$ zeros) for every positive integer $k$.
The difference with $R_{70}$ is in that the top rows (ones that are above the "Toeplitz" part of the infinite matrix) are now allowed to lie in a fixed $F$-algebra, rather than just $F$ (see p22). On p33, the algebra is chosen to be the ring $R_1$ from p20 (the symbol $R$ denotes a different ring in Section 3 from what is taken as $R$ in Section 6). I renamed $R$ to $T$ and $R1$ to $C$.
Ster defines $V$ just as infinite-dimensional, not necessarily countable-dimensional. On the expanded details page for $R{70}$, it is written "infinite" but probably meant "countable".
In $R{67}$ the Bergman's Toeplitz-like ring appears in the "scalars" of a matrix ring direct limit. Here the construction is done vice versa: the "scalars" are subrings of a $\Bbb{CF}{\Bbb N}(F')$, but the "large structure" is Toeplitz-like.
Properties
Each property in my first #21 comment translates to this situation directly, except IBN and vNr:
:x: Dedekind finite. The classical example put into $E_{11}$ with other diagonal elements being the identity transformations works fine.
:x: IC. Same technique as in other known examples should work: take the zero right ideal and the first row for $B$.
:x: countable. The first few rows in the construction can be completely arbitrary.
Maybe someday I'll try to adapt Goodearl p47 to vNr here...
Let $[F':F]=n$ be a field extension, $S \supset S'$ the full $F$- and $F'$-linear rings of the countable-dimensional $F'$-vector space $V$, and their ideals $I \supset I'$ of transformations with finite-dimensional range. Then the sum $T = S' + I$ is a subring in $S$.
The idempotent $e = E_{11}$ of $S'$ is primitive in $S'$ and is a sum of $n$ primitive orthogonal idempotents $e_1, \ldots, e_n$ in $S$ and $T$. Let $C$ be the corner ring $(e_n + 1 - e) T (en + 1 - e)$. Then the ring $R{83}$ is defined as the set of all countable-sized column-finite matrices $M$ over $C$ with the following property: there exists an integer $N = N(M)$ such that the row $M{N,*} = [m{N1}, m{N2},\ldots]$ has all $m{Ni} \in F = F1 \subset C$, and all the next rows are shifts of $M{N,*}$, namely, $M{N + k, *} = [0, \ldots, 0, m{N1}, m{N2},\ldots]$ ($k$ zeros) for every positive integer $k$.
Cf. $R_{67}$ and $R_{70}$.
Explanation
The difference with $R_{70}$ is in that the top rows (ones that are above the "Toeplitz" part of the infinite matrix) are now allowed to lie in a fixed $F$-algebra, rather than just $F$ (see p22). On p33, the algebra is chosen to be the ring $R_1$ from p20 (the symbol $R$ denotes a different ring in Section 3 from what is taken as $R$ in Section 6). I renamed $R$ to $T$ and $R1$ to $C$. Ster defines $V$ just as infinite-dimensional, not necessarily countable-dimensional. On the expanded details page for $R{70}$, it is written "infinite" but probably meant "countable". In $R{67}$ the Bergman's Toeplitz-like ring appears in the "scalars" of a matrix ring direct limit. Here the construction is done vice versa: the "scalars" are subrings of a $\Bbb{CF}{\Bbb N}(F')$, but the "large structure" is Toeplitz-like.
Properties
Each property in my first #21 comment translates to this situation directly, except IBN and vNr:
Maybe someday I'll try to adapt Goodearl p47 to vNr here...
_Originally posted by @dyunov in https://github.com/rschwiebert/dart_data/issues/22#issuecomment-1861217067_