Real closed rings and real closed ∗ rings

Here we will distinguish and compare different notions of real closedness, mainly one developed by N. Schwartz in his Habilitationschrift and the other developed by A. Sankaran and K. Varadarajan in [17], which we shall call real closed ∗. We adhere to the definition of real closed rings as defined and characterized in Schwartz (1986) [18] and we determine and characterize real closed rings that are real closed ∗. We also show that f-rings (and thus also real closed rings) have unique real closure ∗.