Triangular Matrix Representations and Triangular Matrix Extensions

$\pi$-BAER $\ast$-RINGS

“…Let us recall that a * -ring (or an involutive ring) R is a ring with an operation * : R → R, called involution, such that (x + y) * = x * + y * , (xy) * = y * x * , and (x * ) * = x, for all x, y ∈ R. An idempotent p of a * -ring R is called a projection if p is self-adjoint (p * = p). An idempotent e ∈ R is called right (resp., left) semicentral if ex = exe (resp., xe = exe), for all x ∈ R [11]. We denote by S r (R) (resp., S (R)) the set of all right (resp., left) semicentral idempotents of R. If X is a nonempty subset of R, then r R (X) (resp., R (X)) is used for the right (resp., left) annihilator of X over R. We use M n (R), R[x], and R [[x]] for the n by n full matrix ring over R, the ring of polynomials, and the ring of formal power series, respectively.…”