On Left–Right Consistency in Rings

Left-right consistency in rings III

“…This is the analogue of (3.1) obtained by beefing up (3.5) to (5.4); alternatively it is a simple observation [4] that everything in the closure of the invertibles is left-right consistent. Here the closure is of course the norm closure, but the proof is routed through what we call [2] the spectral closure.…”

“…We remark that, in a change from [8], we are following the notation of Cao/Zhang/ Zhang [1]. The characterization [4] of left-right consistent elements can ([7] Theorem 4) be expressed spectrally:…”

“…When K = A −1 we get the "consistently invertible elements" of A. ϖ(K) is always [3], [4] a sub-semigroup of A:…”

“…We recall [5], [3], [4] that an element a ∈ A is said to be "left-right consistent" if for arbitrary b ∈ A the two products ba and ab are invertible or not together:…”