Relating Properties of a Ring and Its Ring of Row and Column Finite Matrices, ,

Mackey and Ornstein proved that if R is a semi-simple ring then the ring of row and column finite matrices over R (RCFMΓ(R)) is a Baer ring for any infinite set Γ. A ring with identity is a Baer ring if every left (equivalent every right) annihilator is generated by an idempotent. This result is discussed in Kaplanskyʹs book, “Rings of Operators.” This result is of course decades old. Here we prove that the converse is true. The proof is long and we develop techniques which allow us to obtain results of a more modern flavor about RCFMΓ(R), where R is a perfect or semi-primary ring. Finally, we obtain good enough results on annihilators in RCFM( ) to show that this ring is coherent.