A closure operation in rings

We study the operation E → cl(E) defined on subsets E of a unital ring R, where cursive Greek chi ∈ cl(E) if (cursive Greek chi + Rb) ∩ E ≠ 0 for each b in R such that Rcursive Greek chi + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl(L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl(Re) = Re + rad(R) if e is an idempotent in R, and cl(I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad(R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl(R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element cursive Greek chi with partial inverse y is persistently regular, if and only if the skew corner (1 - cursive Greek chiy)R(1 - ycursive Greek chi) is contained in Rr. If Ireg(R) denotes the maximal regular ideal in R and R-1q the set of quasi-invertible elements, defined and studied in [4], we prove that R-1q + Ireg(R) ⊂ Rpr. Specializing to C*-algebras we prove that cl(E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, R-1l, R-1r, R-1q and R-1sa , and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.