ON SEPARABLE ABELIAN EXTENSIONS OF RINGS

Let R be a ring with 1, G(=〈ρ1〉×…×〈ρm〉) a finite abelian automorphism group of R of order n where 〈ρi〉 is cyclic of order ni. for some integers n, ni, and m, and C the center of R whose automorphism group induced by G is isomorphic with G. Then an abelian extension R[x1,…,xm] is defined as a generalization of cyclic extensions of rings, and R[x1,…,xm] is an Azumaya algebra over K(=CG={c   in   C/(c)ρi=c   for each   ρi   in   G}) such that R[x1,…,xm]≅RG⊗KC[x1,…,xm] if and only if C is Galois over K with Galois group G (the Kanzaki hypothesis).

[3,4,5], and others.In [3], a separable cyclic extension Rgxl with respect to a cyclic automorphism group ( of R of order n for some integer n over a noncommutative ring R was studied.It was shown (3, Theorem 3.3) that if R is Galois over R ( (= {r in R / (r) r}) with Galois group and if R is contained in the center C of R, then R[x] is an Azumaya algebra over R where x (= b for some b in R) and n are units in R Let G be an abelian automorphism group of R of order n such that G X "Km G. SZETO   for some integers m, and n..
where (i is a cyclic subgroup of order n i Noting that (C)i C for each fi' we shall study an abelian extension n.
R[Xl,...,Xm] with respect to G, where rx i xi(ri) for each r in R, xi k k b. which is a unit in CG, x.x.
x.x. for all i and j, and the set x ...x m / Oki ni is a basis over R. A ring R is called to satisfy the Kanzaki hypothesis ([6], P. 110) if R is Azumaya over C with a finite automorphism group G and C is Galois over K (= C G) with Galois group induced by and isomor- phic with G. DeMeyer [7] has shown that R = R GC under the Kanzaki hypothe- sis for R. The present paper will generalize the Parimula-Sridharan theorem from cyclic extensions ([2], Proposition 1.1, [3], Theorem 3.3) to abelian ex- tensions R[Xl,...,Xm] with respect to an abelian automorphism group G (= ...K(?m>) of R. Let G restricted to C be isomorphic with G. Then we shall show that C is Galois over K (= CG) if and only if R[x 1,...,xm] is an Azumaya algebra over K such that R[x 1,...,xm RG(R)KC[XI'''''Xm where R G is an Azu- maya K-algebra.Thus, a structure of R[Xl,...,Xm] is obtained.Moreover, a structure of C[Xl,...,Xm is also obtained when each direct summand of G is a G-subgroup (see definition below).

PR EL IM INAR I ES.
Throughout, let R be a ring with I, C the center of R, G (= (I>X...X (m>) an abelian automorphism group of R of order n where i is cyclic of order n i for some integers n, ni, and m.Then R[x ,...,Xm is the abelian extension of R with respect to G as defined in Section I. We denote C G by K, and assume that the automorphism group of C is isomorphic with G.The Azumaya algebra R is called to satisfy the Kanzaki hypothesis ([6, P. 110) if C is Galois over K with Galois group induced by and isomorphic with G.

ABELIAN EXTENSIONS.
Keeping the notations of Sections and 2, we shal show the Parimula- Sridharan theorem ([2], Proposition 1.1, [3], Theorem 3.3) and two structural theorems for abelian extensions R[x 1,...,xm.We begin with a proposition on separable abelian extensions.PROPOSITION 3.1.Let G (= (1>X...XKm) be an abelian automorphism ni (- are units in C G for each i, then group of R of order n.If n and x i b i R[x 1,...,xm] is a separable extension of R. is a unit in C G. Hence the cyclic ex- PROOF.Since n i divides n, n i tension R[xI] with respect to 'I is a separable extension over R ([3], Lemma 3.1).Now 2 > is extended to an automorpism group of R[x1 by (xI) ) is a separable extension over R by the transitivity of separable extensions.By repeating the above argument (m-2) times, RXl,...,Xm is a separable extension over R.
THEOREM 3.2.By keeping te notations of Proposition 3.1, if R satis- fies the Kanzaki hypothesis, then R[x 1,...,xm is an Azumaya K-algebra.PROOF.By Proposition 3.1, R[x 1,...,xm is a separable extension over R. By the Kanzaki hypothesis for R, R is separable over C and C is Galois over K, so R[Xl,...,Xml is a separable extension over K by the transitivity of separable extensions.So, it suffices to show that the center of R Ix 1,...,x] is K.It is e.asy to see that K is contained in the center.
Then, similar to the arguments used in the proof of Theorem 3.2, we shall R G show that the commutant of C[x 1,...
is contained in the commutant.Now, let f ao+X ...Xm .abe an element in the commutant for some a and a in R and Okini.Then c f fc for each c in C.This implies that a O. Also, xif fx i for each i, so a is in RG.Thus f (= ao) is in RG.Noting that C[xl,...,Xm and R[xl,...,Xm are Azu- maya algebras over K, we have that R[x 1,...,xm RGKC[X1,...,xm by the well known commutant theorem for Azumaya algebras ([7, Theorem 4.3, P. 57).
COROLLARY 3.4.If R satisfies the Kanzaki hypothesis, then R G is an Azumaya algebra over K.
PROOF.This is a consequence of Theorem 3.3 and the commutant theorem for Azumaya algebras.
We are going to show a converse of Theorem 3.3.THEOREM 3.5.If R[x 1,...,xm is an Azumaya algebra over K such that N RGK C R G Rx 1,...,xm Ix|,.. ,Xm where is an Azumaya K-algebra, the C is Galois over K with Galois group induced and isomorphic with G. PROOF.By the commutant theorem for Azumaya algebras, since R[x1,...,Xm] and R G are Azumaya K-algebras, so is C[Xl,...,Xm.Then, we claim that C is Galois over K with Galois group G. Suppose not.There is a non-identity g in G such that {c-(c)g / c in C is not C ([7], Proposition 1.2).Let g k k I "''m m for some ki, Oki<ni.Since I generated by (c-(c)g) for c in C is a G-ideal of C (that is, (I)G I), we have an Azumaya algebra (C/l)[Xk 1''''Xm over K/(KI).On the other hand, one can show that (x11"''Xmm) is in the center of (C/I)[xl,...,Xm.This is a eontradition.
Thus C is Galois over K with Galois group G.
Let S be a ring Galois extension over a subring T with a finite Galois group G.A normal subgroup H of G is called a G-subgroup if S is Galois over S H with Galois group H and S H is Galois over T with Galois group G/H.Keep- ing the notations of Theorem 3.5, we give a structural theorem for C[xI,...,x for each i We denote the center of C[x 1,...,xi_ 1,xi+1,...,xm] by C i (/i )
As immediate consequences of Theorem 3. PROOF.Since C is Galois ever K with no idempotents but 0 and I, each direct summand of G is indeed a G-subgroup ([7], Theorem 1.1, P. 80, or [8).
the center of R[x ,...,x such that f a +x ...x .awhere a m o m oand a are in R, and Oki< n..Then, rf fr for each r in R.This implies m that ra aor and ar (r)1.a.Hence a is In C, and the second k k a(r-(r)11...mm )_ 0 for each r in C. Thus a is inequation implies that k k r-(r)11...mm / r in C of R. Since R is Ann- the annihilator ideal I of k kis the annihilator ideal of --r-(r) of C. I {0([7], Proposition 1.2) because C is Galois over K with Galols group induced by and isomorphic with G. i.Thus a is in K.This completes the proof.o Next is a structural theorem for R[Xl,...,xm] under the Kanzaki hypo- thesis.THEOREM 3-3.If R satisfies the Kanzaki hypothesis, then R[x 1,...,xm RGC[Xl,...,Xml as Azumaya K-algebras.

8 .
5 and Theorem 3.6, we have the following: COROLLARY 3.7.If R satisfies the Kanzaki hypothesis such that each di- RGK 11% m rect summand of G is a G-subgroup, then R[x 1,...,xm] If R satisfies the Kanzaki hypothesis such that the center C of R has no idompotents but 0 and I, then R[x 1,...,xm RGKC [x11 K'" "KCz c Xm]"