On automorphism group of free quadratic extensions over a ring

Let R be a ring with 1 , ρ an automorphism of R of order 2 . Then a normal extension of the free quadratic extension R [ x , ρ ] with a basis { 1 , x } over R with an R -automorphism group G is characterized in terms of the element ( x − ( x ) α ) for α in G . It is also shown by a different method from the one given by Nagahara that the order of G of a Galois extension R [ x , ρ ] over R with Galois group G is a unit in R . When 2 is not a zero divisor, more properties of R [ x , ρ ] are derived.