Relating properties of a ring and its ring of row and column finite matrices. (English) Zbl 0991.16025
The Mackey-Ornstein theorem, mentioned in I. Kaplansky’s book “Rings of operators” (1968; Zbl 0174.18503), states that if \(R\) is a semisimple ring then the ring \(\text{RCFM}_\Gamma(R)\) of row and column finite matrices over \(R\) is a Baer ring for any infinite set \(\Gamma\). Here a ring with identity is a Baer ring if every left (equivalently every right) annihilator is generated by an idempotent. The authors prove that the converse is true, i.e., \(R\) must be semisimple if \(\text{RCFM}_\Gamma(R)\) is a Baer ring for some infinite set \(\Gamma\). The proof is long and the authors develop techniques to obtain more results for \(\text{RCFM}_\Gamma(R)\), where \(R\) is a perfect or semiprimary ring. Finally, they obtain results on annihilators in \(\text{RCFM}_\mathbb{N}(\mathbb{Z})\) to show that this ring is left and right coherent.
Reviewer: Xue Weimin (Fujian)
MSC:
| 16S50 | Endomorphism rings; matrix rings |
| 16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
| 16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |
Keywords:
rings of row and column finite matrices; semisimple rings; Baer rings; idempotents; annihilators; coherent ringsCitations:
Zbl 0174.18503References:
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