The ring of infinite matrices over $\Bbb Q$ with entries indexed by positive integers such that every row and every column contains only a finite number of nonzero entries.
This is contained in $R_{15}$ and contains the ring of finite matrices (which is TBA), but also every countably-generated $\Bbb Q$-algebra, see P. Nielsen "Row and column finite matrices" Cor6.
References
[1] Camillo, Costa-Cano, Simon "Relating Properties of a Ring and Its Ring of Row and Column Finite Matrices".
[2] Simon "Finitely Generated Projective Modules over Row and Column Finite Matrix Rings".
[3] O'Meara "The exchange property for row and column-finite matrix rings".
[4] Ara, Pedersen, Perera "An infinite analogue of rings with stable range one".
Properties
(strikethrough = follows from ones already added in #68)
[x] ✔ exchange. [3] Th1.
- [ ] ❌ Dedekind finite. The "infinite Jordan blocks" are RCFM.- [ ] ✔ prime. Lam II section 14F p401.
[x] ❌ IBN.
- [ ] ❌ polynomial identity. (Neither of the four rings are.)
[x] ❌ von Neumann regular [3] introduction
Remarks
[4] 8.8(A) p36: the rng of finite matrices is the only nontrivial ideal of RCFM. They also claim that the non-finite idempotents are all equivalent to 1.
Description
This is contained in $R_{15}$ and contains the ring of finite matrices (which is TBA), but also every countably-generated $\Bbb Q$-algebra, see P. Nielsen "Row and column finite matrices" Cor6.
References
[1] Camillo, Costa-Cano, Simon "Relating Properties of a Ring and Its Ring of Row and Column Finite Matrices". [2] Simon "Finitely Generated Projective Modules over Row and Column Finite Matrix Rings". [3] O'Meara "The exchange property for row and column-finite matrix rings". [4] Ara, Pedersen, Perera "An infinite analogue of rings with stable range one".
Properties
(strikethrough = follows from ones already added in #68)
- [ ] ✔ semiprimitive. [2] Theorem 4.5.- [ ] ❌ Dedekind finite. The "infinite Jordan blocks" are RCFM.- [ ] ✔ prime. Lam II section 14F p401.- [ ] ❌ polynomial identity. (Neither of the four rings are.)Remarks