ON SEPARABLE EXTENSIONS OF GROUP RINGS AND QUATERNION RINGS

lqe purposes of the present paper are (i) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extension RG (R may be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extension RQ over a ring R, where Q are the usual quaternions i,j,k and multiplication and addition are defined as quaternion algebras over a field. We shall show that RG has a unique separable idempotent if and only if G is abelian, that there are more than one separable idempotents for a separable quaternion ring RQ, and that RQ is separable if and only if 2 is invertible in No


INTRODUCTION.
M. Auslander and O. Goldman ([I] and [2]) studied separable alge- bras over a commutative ring.Subsequently, the investigation of se- parable algebras (in particular, Brauer groups and Azumaya algebras) has attracted a lot of researchers, and rich results have been obtained (see References).K. Hirata and K. Sugano ( [5]) generalized the con- cept of separable algebras to separable ring extensions; that is, let S be a subring of a ring T with the same identity.Then T is called a separable ring extension of S if there exists an element aibi in TST such that x([aibi) ([aibi)x for each x in T and [aib i I.
Such an element aibi is called a separable idempotent for T. We note that a separable idempotent takes an important role in many theo- reins (for example, see [6], Section 5,6, and 7).It is easy to verify that (I/n)(gig1) and ei1eli ([4], Examples II and III, P. 41) are separable idempotents for a group algebra RG and a matrix ring M (R) m { } with n invertible in R and e.. are respectively, where G gl,...,g n m3 matrix units.We also note that the separable idempotent for a commu- tative separable algebra is unique ([6], Section I, P. 722).

PRELIMINARIES.
Throughout, G is a group of order n, R is a ring with an identity I.The group ring RG rig i / r i in R and gi in G}, which is a free R-module with a basis {gi and (rigi)(sigi) tkg k where t k ris j for all possible i,j such that gig j gk" The ring R is imbed- ded in RG by r-rgl, where gl is the identity of G (gl I).The mul- tiplication map RGRRG--RG is denoted by .Clearly, {gigj / i,j ,... ,n form a basis for RGRRG.An element rij in RGRRG is called a commutant element in RGRRG if x(rij (gigj)) (rij(gigj))x for all x in RG.

MAIN THEOREMS.
We begin with a representation for (x) for a commutant element x in RGRRG and then we show that RG has a unique separable idempotent if and only if G is abelian.LEMMA I. Let x ri.i (gig.i )' i,j 1,...,n, be a commutant e!e- ment in RGRRG.Then (x) i=ml .rlk: )nk.Ck.where m is the number of conjugate classes of G, nk. is the order of the normalizer of gk.' and C k is the sum of different conjugate elements of --., for some k.PROOF.Since x is a commutant element, xg for each gp P gpgk x is rlk, and the coefficient of The coefficient of the term in gp gp gqgp gk" Hence rlk rpq whenever the same term in x is rpq, where gqgp gk" Thus x krlk (gpgq), where p,q run over 1,...,n, such -I).TakingS(x) that gqgp gk; that is, x krlk(pgpgkgp -I nkCk where n k is the -I).For a fixed k pgpgkgp order of the normalizer of S k and C k is the sum of all different conju- gate elements of gk" Hence ,(x) k=]rknkCk.Renumerating elements we let [gkl''''gkm } be all non-conjugate ele- ments of each other; then {C k ,...,C k } are all different elements in m m the set, {CI,...,Cn}.Thus (x) i:i (rlk :)nk.Ck.where rlk are coefficients of the same Ck., and m is the number of conjugate classes of G.
THEOREM 2. Let RG be a separable extension of R.Then, RG has a unique separable idempotent if and only if @ is abelian.
PROOF.Let x Zrij(gig j) be a separable idempotent for RO.Ckl and n kl n, the order of G. Since (x) 1, (rlkl)nkl Ckl and (rlk)n k and (rlk,)n k C k 0 for each i 1. Toting kl i i i that Ckl I, we have rlk r11 and so the first equation becomes rlln .Hence the order of G, n, is invertible in R. Thus nk., bemng a factor of n, is also invertible in R.But conjugate classes form a partition of G, so (rlk,)nk.C,.. 0 implies that rlk 0 for each I 1 I i i I.This system of homogeneous equations rlk 0 in the unknowns 1 rlk with i 6 has trivial solutions if and only if n m, and this holds if and only if G is abelian.Since the uniqueness of the separable idempotent (= (I/n)(gig1)) is equivalent to the existence of trivial solutions of the above system of equations, the same fact is equi- valent to G being abelian.
The theorem tells us that there are many separable idempotents for a separable group ring RG when G is non-abelian.Also, we remark that if RG is a separable extension of R, the order of G is invertible in R from the proof of the theorem.Next, we discuss another popular separ- able ring extension, a quaternion ring extension RQ, where RQ (r1+rii+rjj+rkk / i,j, and k are usual quaternions). (RQ,+.) is a ring extension of R under the usual addition and multiplication similar to quaternion algebras over a field.Now we characterize a separable idem- potent for a separable quaternion ring extension RQ.THEOREM 3. Let RQ be a separable quaternion ring extension.Then a commutant element x rst(St), s,t 1,i,j,k, in RQRR Q is a separ- able idempotent for RQ if and only if r11 I/4.PROOF.Since x is a commutant element in RQRQ, ix xi.The co- efficients of the term 191 on both sides are -ril and -rli so ril rli.
Since jx xj, the coefficients of the term k1 on both sides are -ril r Also, kx xk, so the coefficients of the term j1 -rkj so ril k0 on both sides are -ril rjk.Hence rli ril rkj -rjk.Similarly, by comparing coefficients of other terms, we have r11 -rii -rjj -rkk rlj rjl -rki rik and rlk rkl -rij rji.In other words, rst rpq if ts qp, and rst -rpq if ts -qp.Thus r114+rliO+rljO+rlkO 4r11.Consequently, x is a se- parable idempotent if and only if r11 I/4 (for (x) I).
COROLLARY 4. Let RQ be a quaternion ring extension of R. Then RQ is separable if and only ,if 2 is invertible in R.
PROOF.The necessity is immediate from the theorem.The sufficien- cy is clear since the element x with r11 I/4, rli rlj rlk 0 as given in (*) in Theorem 3 is a separable idempotent for RQ.
REMARK.It is easy to see that every x of the form (*) in Theorem 3 with r11 rli rlj and rlk in the center of R is a commutant element in RQRR Q.Hence, from the proof of Theorem 3, the complete set of corn- -ts}.Also, the complete set of separable idempotents for Pq RQ is a subset of C such that r 11 I/4 and r Ii' rlj' rlk center of R. Thus there are many separable idempotents.
are in the of g.. Let gk I, the identity of G. Then m x r (11-ii-jj-kk)+rli( 1i+i1-jk+kj)+rlj(1j+j1-ki+ik)+ r Ik 1k+k -ij+ ji * But then (x) Since conjugate class- es form a partition of G, C i C. if and only if gi is conjugate to gj.