On automorphism group of free quadratic extensions over a ring (Q1061206)

Let R be a ring with 1, and \(\rho\) an automorphism of R of order 2. A free quadratic extension with respect to \(\rho\) is any ring with a free basis \(\{\) 1,x\(\}\) over R such that \(rx=x(r\rho)\), \(x^ 2=b\) which is a unit in the center of R such that \(b\rho =b\). An R-automorphism \(\alpha\) of R[x,\(\rho\) ] is a ring automorphism such that \((r+xt)\alpha =r+(x\alpha)t\) for r,t\(\in R\). A ring T is called a normal extension of a subring S with respect to an automorphism group G of T if \(T^ G=S\), where \(T^ G=\{t\in T|\) \(t\alpha =t\) for each \(\alpha\in G\}\). A ring T is called a Galois extension over S with a finite Galois group G if T is normal over S and there are \(a_ i,b_ i\in T\), \(i=1,2,...,n\), such that \(\sum_{i}a_ ib_ i=1\) and \(\sum_{i}a_ i(b_ i\alpha)=0\) for each \(\alpha\) \(\neq 1\) in G. A normal extension R[x,\(\rho\) ] is characterized in terms of the elements x-x\(\alpha\), where \(\alpha\) are R-automorphisms of R[x,\(\rho\) ]. This enables a different method from the one given by Nagahara to show that 2 is a unit in R if R[x,\(\rho\) ] is Galois over R. Theorem. Assume 2 is not a zero divisor in R. If R[x,\(\rho\) ] is a normal extension over R with respect to a cyclic R-automorphism group \(<\alpha >\), then x-x\(\alpha\) is not a zero divisor in R[x,\(\rho\) ]. All R- automorphism groups of R[x,\(\rho\) ] of order 2 such that R[x,\(\rho\) ] is a normal extension over R are determined. Three examples are given to illustrate the results of the paper.