ON GENERALIZED QUATERNION ALGEBRAS

Let B be a connutative ring with i, and G (={o}) an automorphlsm group of B of order 2. The generalized quaternion ring extension B[j over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free 2 B-odule with a basis {l,j}, and (2) j -l and jb o(b)j for each b in B. The purpose of this paper is to study the separability of B[j ]. The separable extension of B[j] over B is characterized in terms of the trace (= i+o) of B over the subrlng of fixed elements under o. Also, the characterization of a Galols extension of a connutative ring given by Parlmala and Sridharan is improved. KEF WORDS AND PHRASES. Quaternion Rings, Separable Algebras, and alois AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES. 16A16, 13A20, 13B05.


I. INTRODU CTI
In 6, we studied the separable extension of group rings RG and quaternion rings Ri,j,k over a ring R with I. We have shown that Ri,j,k is a separable extension of R if and only if 2 is a unit in R.
Recently, S. Parimala and R. Sridharan (5) investigated another class of quaternion ring extensions Bj over a commutative ring B with and with an automorphism group G (= ) of order 2, where Bj is a free B-module with a basis 1,j), j2 -I, and jb 6(b)j for each b in B. Their work is based on the following characterization of a Galois extension of a commutative ring (5, Proposition 1.1): Let A be the set of elements in B fixed under .Assume 2 is a unit in A. Then, B is Galois over A if and only if BABJ M2(B), a matrix algebra over B of order 2, where the Galois extension is in the sense of Chase-Har- rison-Rosenberg (2).The purpose of this paper is to study the separ- ability of Bj.Without the assumption that 2 is a unit in A, we shall characterize the separability of Bj in terms of the trace (= I+) of B over A. This shows the existence of a separable generalized quaternion ring extension B with 2 not a unit in A. When Char(A) 2, we shall show that Bj is a separable extension over B if and only if B is Galois over A. Thus we can improve the above theorem of Parimala and Sridharan.Then, the case in which 2 is a unit will be discussed, and several examples are constructed to illustrate our main results.

PRELIMINARIES.
Let us recall some basic definitions as given in I,2,3,4 and 6.Let B be a commutative ring containing a subring A with the same identity I. Then B is called a Galois extension over A (2, or 31, Chapter 3) with a finite automorphism group G if (I) there exist elements ai,b i in B / i 1,2,...,n for some integer nsuch that aibi and ai6(bi) 0 whenever 6 in G, and (2) A b in B / (b) b for all 6 in G.The map 6 is called the trace of B over A denoted by Tr.Let S be a ring (not necessarily commutative) contain- ing a subring R with the same identity I. Then S is called a separable e_xtension of R if there exist elements, (ci,d i in S / i 1,2,...,n for some integer n} such that (I) a(ci(R)di) (cidi)a for all a in S where @ is over R, and (2) cidi I.Such an element cidi is called a separable idempotent for S. When R is contained in the center of S, S is called a separable R-algebra.The separable R-algebra S is called an Azumaya R-alge.braif R is the center of S.
Throughout, we assume that B is a commutative ring with I, and G (= {6) an automorphism group of order 2 of B, and that Bj3 is the generalized quaternion algebra over A, where A is the subring of elements fixed under 4. Our main goal in the section is to study a separable ex- tension Bj3 over B without the assumption that 2 is a unit in A. We begin with a description of the set of separable idempotents for B[j3 (if there are any) over B. Clearly, 11,1j,j1,jj} is a basis for LEMMA 3.1.The element x a11(1(R)1)+a12(1j)+a21(J(R)1)+a22(J(R) j) is a separable idempotent for B[j] over B if and only if (I) a22 -6(a11) such that Tr(a11) I, and (2) a21 6(a12) such that a12((b-6(b)) 0 for all b in B and Tr(a12) O.
Using Theorem 3.2, we can obtain a characterization of a separable extension Bj over B when Char(A) 2.
THEOREM 3.3.Assume Char(A) 2.Then, Bj is a separable extension over B if and only if B is a Galois extension over A.
PROOF.Let B be a Galois extension over A. Corollary 1.3 on P. 85 in 3 implies that Tr(c) for some c in B. Thus Bj is a separable extension over B by Theorem 3. Let us recall that the theorem of Parimala and Sridharan (Propo- sition 1.1 in 5): Assume 2 is a unit in A. Then, B is Galois over A if and only if BAB[J M2(B), a matrix algebra over B of order 2.
We are going to improve it without the assumption that 2 is a unit in A. THEOREM 3.4.If B is Galois over A, then BABJ M2(B ).
PROOF.If B is Galois over A, there exists an c in B such that Tr(c) (3, Corollary 1.3, P. 85).Hence BKjl is a separable ex- tension over A by Theorem 3.2.But B is also a separable extension over A by Proposition 1.2 in 3], so the transitive property of separ- able extensions ([4], Proposition 2.5) implies that Bj is a separable A-algebra.Moreover, we claim that (I) B[j] is an Azumaya algebra over A, and (2) B is a maximal commutative subalgebra of Bj.The proof of these facts was given in [7].For completeness, we give an outline here.For part (I), it suffices to show that A is the center of Bj].Clearly, A is contained in the center.Now, let b+b'j be in the center.Then j(b+b'j) (b+b,j)j and c(b+b'j) (b+b'j)c for each c in B. Equating coefficients of the basis 1,j} in the above equations, we have that b is in A and b' 0 by Statement 5 in Proposition 1.2 on P. 81 in 3].For part (2), to show that B is a maximal commutative The computation is similar to part (I).
Moreover, noting that B is separable over A, we then conclude that BA(BjS) HomB(BjS,Bj] by Theorem 5.5 on P. 65 in 3, and this implies that BABJ] _ M2(B), where (BjS) is the opposite ring.
In 7], the sufficiency of the Parimala and Sridharan theorem was shown by a different method from [5.Now we slightly improve the statement without the assumption that 2 is a unit in A.
THEOREM 3.5.Let B[j] be a separable extension over B. If BABJ] M2(B), then B is Galois over A.
PROOF.Since BCj3 is a separable extension over B, there exists an element c in B such that Tr(c) by Theorem 3.2.Hence the se- quence B-*A-0 is exact under the trace map.But A is projective over A, so the sequence splits, and then A is an A-direct summand of B. By hypothesis, BAB[J M2(B which is an Azumaya B-algebra, so BCj is an Azumaya A-algebra (3], Corollary 1.10, P. 45).Therefore B is Galois over A by using the same argument as given in 7.
. .PEGIL PBLE UAT.RNION ALGEBS.Theorem 3.5 tells us that B is an Azumaya A-algebra such that M2(B) when B is Galois over A. In this section, we are go- ing to discuss generalized quaternion algebras Bj in which 2 is a unit in A when B is projective and separable over A. With a similar argument as given in Lemma 3. I, we have M 4.1.The element a11(11)+a12(1j)+a21(J1)+a22(jj) in A]AAJ] is a separable idempotent for Aj3 if and only if (I) a22 -a11 such that 2a 11 I, and (2) a21 a12 such that 2a 12 0.
THE0M 4.2.The A-algebra A[j] is separable if and only if 2 is a unit in A. PROOF.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: 2. Conversely, by Theorem 3.2 again, there exists an c in B such that Tr(c) I, so (c+(c)) I.By hypo- thesis, Char(A) 2, (c) (-c) =-(c), so c-(c) I. Hence the ideal generated by (b-(b)) / b in B)= B. This implies that B is Galois over A by the statement 5 in Proposition 1.2 on P. 81 in 3.

First
Round of Reviews March 1, 2009 The necessity is clear by Lemma 4.1; the sufficiency is immediate because (I/2) (11-jj) is a separable idempotent.Bj] is a separable extension over B and projective over Aj as a bi- module if and only if 2 is a unit in A.