On separable cyclic extensions of rings

Abstract

The quaternion algebra of degree 2 over a commutative ring as defined by S. Parimala and R. Sridharan is generalized to a separable cyclic extension B\J] of degree n over a noncommutative ring B. A characterization of such an extension is given, and a relation between Azumaya algebras and Galois extensions for B[j] is also obtained.

On separable noncyclic extensions of rings

“…= (2 a, ® i,-)f for each t in 7? and 2 a,Z>, = 1 where ® is over S (Szeto and Wong (1982)). Such an element 2 a, ® b t is called a separable idempotent for /?…”

“…Let R be a ring with 1 (not necessarily commutative), p an automorphism of order n of R for some integer n. The separability of the cyclic extension R [i, p], has been intensively investigated (Parimula and Sridharan (1977), Nagahara and Kishimoto (1978), Szeto (1980), Szeto and Wong (1982)), where ri = i(r)p for each r in R, {1, i, i 2 ,...,/""'} is a free basis of R [i, p] over R, i" = b which is a unit in the center C of R and (b)p -b. The purpose of the present paper is to continue the above investigation to a noncyclic extension: a crossed product A (R,G), where G is a finite automorphism group (not necessarily cyclic) with factor set /: GX G -> U{C), the set of units of the center C of R. Our study includes cyclic extensions, crossed products over a commutative ring (DeMeyer and Ingraham (1971), Chapter 3), and crossed products with trivial factor set (Kanzaki (1964), Section 3).…”