New characterizations of von Neumann regular rings and a conjecture of Shamsuddin (Q677839)

The author proves two theorems and a conjecture of Shamsuddin. Let \(R\) be a ring with Jacobson radical \(J\). Then (1) \(R\) is von Neumann regular (=VNR) if and only if \(R\) is semiprime, \(R/J\) is VNR, the maximal von Neumann regular ideal \(M(R)\) splits off, and \(J\) is an annihilator (left or right) ideal. (2) If \(R\) is two-sided continuous, then \(R\) is VNR if and only if \(R\) is a semiprime ring and \(J\) is an annihilator ideal. (3) If \(R/J\) is VNR and a finite product of simple ideals \(A_1,\dots,A_n\) then \(R\) is VNR if and only if \(R\) is a semiprime ring and \(J\) is an annihilator ideal.