Abstract The aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear 539:117–133, 2018) by a unified approach. We show general theorem with respect hereditary property. Then vast majority decompositions known algebra Hilbert space operators generalized elements Baer $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/M...

It was pointed out in my last paper that there are rings whose real closure ∗ are not unique. In [4] we also discussed some example of rings by which there is a unique real closure ∗ (mainly the real closed rings). Now we want to determine more classes of rings by which real closure ∗ is unique. The main results involve characterisations of domains and Baer regular rings having unique real clos...