  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 closure *, and an example showing that regular rings need not be
f-rings in order to have a unique real closure *. The main objective here is to
find characterisation for uniqueness of real closure * for real regular rings
that will primarily only require information of the prime spectrum and the real
spectrum of the ring.
