Von Neumann regular rings with only finitely many symmetric idempotents (Q1057346)

Let R be a ring with involution \({}^*\). An element e of R is called a projection if \(e^*=e\) and \(e^ 2=e\). The main result of this paper is the following theorem: Let R be a regular ring with involution \({}^*\). Then R has only finitely many projections if and only if R is a direct sum of rings of the following types: 1. a division ring, 2. the direct sum of a division ring D and its opposite \(D^{op}\) with \((a,b)^*=(b,a)\), 3. the \(2\times 2\) matrix ring \(M_ 2(K)\) over a field K with symplectic involution, 4. a finite dimensional matrix ring \(M_ n(F)\) over a finite field and 5. the direct sum of a finite dimensional matrix ring \(M_ n(F)\) over a finite field F and its opposite \(M_ n(F)^{op}\) with \((a,b)^*=(b,a)\).