A characterization of a cyclic Galois extension of commutative rings

On free quadratic extensions of rings

“…Let A = {b in B[ 9 (b) = b} ; they showed ( [4], Proposition 1.1) that B is Galois over A if and only if B @ A B [/] is a matrix algebra over B of order 2. The present authors have generalized the above theorem of PARIMALA and Se,~rtAe, AN from the theory of Azumaya algebras ( [5], Theorem 3.3, and [6], Theorems 3.2 and 3.3). Noting that the ring B0] is a free quadratic extension over B ( [3], [4]), we shall continue our generalizations in two directions: (1) the ring B may be a non-commutative ring, and (2)j 2 may be any invertible element in B.…”

On separable noncyclic extensions of rings

“…We conclude the paper with an example to demonstrate our results. Let R[i, p] be a generalized quaternion algebra (Parimula and Sridharan (1977), Szeto (1980)), where {1, /} is a basis for R [i, p] over R, p an automorphism of R of order 2, ri -i(rp) for each r in R and i 2 = ft in £/(C). We define /: <p>X <p>-» [/(C) by f(p°, p) = f( P , p°) = /(p°, p°) = 1 and /(p, p) = ft. Then it is easy to see that / is a factor set for the crossed product A(R, ( p » with basis U o = U p o, U x = U p such that the identity is U o and that R [i, p] is isomorphic with A(/?,(p» with factor set / under a:…”

“…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).…”

On separable cyclic extensions of rings

“…studied a quaternion algebra B [j] (a free ring extension of degree 2) over a commutative ring B with 1, where j 2 --1 , p(a)j = ja for each a in B and p is an automorphism of B of order 2. One of the present authors (Szeto (1980)) generalized such a ring extension B [j] from degree 2 to a cyclic extension of any degree n, and from a commutative ring B to a noncommutative ring B (Szeto and Wong (to appear)). It was shown that if B is a commutative Galois extension over A, the subring of B consisting of all elements fixed by p, then B [j] is an Azumaya A -algebra (Szeto (1980), Theorem 3.2 and Lemma 3.4).…”

“…One of the present authors (Szeto (1980)) generalized such a ring extension B [j] from degree 2 to a cyclic extension of any degree n, and from a commutative ring B to a noncommutative ring B (Szeto and Wong (to appear)). It was shown that if B is a commutative Galois extension over A, the subring of B consisting of all elements fixed by p, then B [j] is an Azumaya A -algebra (Szeto (1980), Theorem 3.2 and Lemma 3.4). In the present paper, for B [j] over a not necessarily commutative ring B, we shall give a necessary and sufficient condition for the separability of B [j] of degree n over B in terms of the elements of B, and generalize the above result to a noncommutative Galois extension B over A.…”