Ixyl} = {Xly} = {Lxy}, (5) and Such That These Remain Valid Under Extensions of the Ring of Scalars

A GENERAL THEORY OF JORDAN RINGS* BY KEVIN MCCRIMMON

DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE Communicated by N. Jacobson, August 10, 1966 It has long been felt that the definition of Jordan algebras in terms of the Jordan identity [x2,y,x] = 0 (1) is unsuitable for algebras over fields of characteristic 2 or for rings in general; for example, in characteristic 2, it includes the "wrong" algebras (nodal algebras) and excludes the "right" algebras (Jordan matrix algebras). In this note we propose a different definition based on the cubic composition Uy = 2x (x y) -X2y instead of the bilinear composition x y. Our goal is to show that this definition provides a better generalization of the classical theory, and to indicate for the resulting class of algebras a structure theory analogous to that of Professor N. Jacobson.' (A full exposition will appear elsewhere.) 1. Definitions.-Throughout this note, 4b will denote a commutative associative ring with identity element, and I a 4)-module. A quadratic representation on X is a quadratic mapping U: x -k Ux of I into Hom(1X,). If we set Uv = U+- U - U, the cubic composition Uxy induces a trilinear composition Ixyz} = UZ y which is symmetric in x and z. A Jordan algebra with identity element 21 = (X, U,1) consists of a quadratic representation U on X and an element 1 e X which satisfy Uu(x)y = UxUVUx ("fundamental formula"), (2) UA{yxz} = { Uxy z x} ("isotope formula"), (3) U, = 1, (4) Ixyl} = {xly} = {lxy}, (5) and such that these remain valid under extensions of the ring of scalars. This latter condition is equivalent to the assumption that the linearizations of (2) and (3) hold: UU(zvU(,Z~)= UXU1UXZ + UXZUYUX, (2') UU(X)yU(Z), + Uu(XZ)1 = UZUyUz + UZUyUZ + UZZUvUXZy (2") UX{ywz} + Uxt,{yxz} = {U.y z w} + IUx.,y zX}. (3') For convenience, we call an algebra 2l = (X, ) consisting of a symmetric bilinear multiplication x * y on X satisfying the Jordan identity (1) a classical Jordan algebra. The existence of an identity element allows us to introduce a quadratic composition x2 = Ux1. This in turn induces a symmetric bilinear multiplication xOy = Ux ,1 = {xly}. By (5), left (= right) multiplication by an element x is given by Va = Ux,i. A Jordan algebra is a subspace of a Jordan algebra with identity which is closed under the quadratic and cubic compositions. The axiomatic definition of Jordan algebras in terms of the compositions x2 and Uxy seems to be rather involved, since there are many identities relating the two. Important examples are 1072 Downloaded by guest on September 29, 2021 VOL. 56, 1966 MATHEMATICS: K. McCRIMMON 1073 VXUX= Us2= UXVXI (6) 2UX = V_-2, (7) U 2= UX2 (8) Uz0o + UX2,y2 = UXUY+ UYU2+ UX, (9) Ux0o + Uu(c)vY = UXUY + UYUx + VXUYVZx (10) VU(X)y + UX2, + 2Uzoy = UXVY + vYUX + VXVYVX2 (11) (Uxy)2 = UXU1X2, (12) (X2)2= UxX2, (13) {XXy} = x2Oy, and (14)

{xyz} + {yxz} = (xOy)Oz. (15) A quadratic ideal in a Jordan algebra 21 is a subspace e3 such that U.J21 C QA; an ideal is a subspace e3 such that Uz2f c A, UQZ C _S, Q32 C QA, 21O03 C 3. The usual categorical paraphernalia can be introduced. 2. Examples.-The first theorem shows that our definition reduces to the classical one for algebras over fields of characteristic # 2. JUSTIFICATION THEOREM. If there is a scalar 1/2 E b, then there is a natural isomorphism between the categories of Jordan algebras 21 = (1, U,1) with identities and classical Jordan algebras 21 = (X,*) with identities by means of the relations X-Y = '/2xOy 1 = 1 Uxy = 2x.(x y) - (xex)-y. The essence of the proof is contained in relations (6) and (7). Further justification for our definition comes from the fact that the basic Jordan algebras of the classical theory have analogues in the general theory. If S is an associative algebra, we can introduce a Jordan structure G+ on CS by X2 = xx Usy = xyx. A Jordan algebra is said to be special if it is isomorphic to a (Jordan) subalgebra of some (+ for S associative; if it is not special, it is exceptional. For example, if S has an involution*, then the set t(2,*) of *-symmetric elements forms a special Jordan algebra [this includes all the special Jordan matrix algebras ID(Z.,,y) ]. For another example, if Q is a quadratic form on X and 1 E X a point where Q(1) = 1, then the Jordan algebra 21(Q,1) of the quadratic form Q and identity 1 is a special Jordan algebra whose quadratic representation is U-y = Q(x,y*)x - Q(x)y*, (16) where Q(x,y) = Q(x + y) - Q(x) - Q(y) is the linearized form and y -- y* is the involution y* = T(y)1 - y, T(y) = Q(y,1). Here we may take ( to be the Clifford algebra of Q and 1. We obtain a special universal envelope su(21) for a Jordan algebra 21 = (E, U,1) by taking su(21) = Z(X)/ft, where £(X) is the tensor algebra £(X) = X ED X ED ... on X and S is the ideal generated by all lx - x and x y x - Uxy for x,y CEX. Downloaded by guest on September 29, 2021 1074 MATHEMATICS: K. McCRIMMON PROC. N. A. S.

This is an associative algebra with identity having the usual universal property for associative specializattions, so 21 is special if and only if the canonical map su: 21 -- su(21) is injective. [Notice that for 21 = 21(Q,1) the above su(21) is precisely the Clifford algebra of Q and 1. ] 3. Peirce Decompositions.-There is no natural bilinear multiplication between powers of a single element yielding the power-associativity condition Xn Xm = Xn+m (note XnOxm = 2xn+m; here *° = 1, xi = x, Xn+2 = Uzxn), but this is imma- terial to our theory. An element e of a Jordan algebra is idempotent if e2 = e; (13) shows en = e for all n ) 1. Two idempotents ef are orthogonal if Uj = eOf = Ufe = 0. We have the usual Peirce theorems. PEIRCE DECOMPOSITION FOR A SINGLE IDEMPOTENT. If e is an idempotent in a Jordan algebra 21, then 21=21 211/2ED1 (17) is a decomposition of 21 into a direct sum of Peirce spaces 2i = 21i(e) where 21(e) = Ue21 2112(e) ={ Ve- 2Ue} 21, 2(e) = {1 + Ue - Ve} 2. The subspates 21o, W1 are quadratic ideals and for i = 0,1 and j = 1 - i U&2[i C 21i U2111221 C 21i U21/211,2 C 211/2 21,2 C 2li 1/212 C 21 + w2 2i0211/2 C 21/2 { W1/22l/221il C wi {1i%211i/2} C 211/2 {1o211/22} C 211/2, while all other products vanish. Quadratic ideals of the form 21o(e) or 21i(e) will be called Peirce quadratic ideals. PEIRCE DECOMPOSITION FOR A SUM OF IDEMPOTENTS. If 1 = 2ei is a sum of pairwise orthogonal idempotents, then 21 = (E321, (18) is a decomposition of 2t into a direct sum of Peirce spaces 21,j = EiA = 2fji where Eii = Uei, Eij = Ue,ei. The composition rules for distinct indices i,j,k,l are

Uti~iis21, w C 21j, U i C 2[ ii 212 C 2f !tj2 C 21 + 21jj 2iiO21ij C tij j21ijo21, C 21,i, (19) and all other products are zero. A trilinear product of Peirce elements is zero unless the indices can be linked to form a product {ajbjkCktl (i,j,k,l not necessarily distinct). We have { ij2j,,,kll} C 21i, and for distinct i,j,k,l the Peirce relations aijbjjcji = Uei{ (aijObjj)OCji} = Ue,{aijO(bjjOcji)} iibijcjj} = (afiObij)Ocjj = a2jO(bijOcjj) { aibijcj} = UeiA (aijobij) Ocji} { ajbibcjk = (a11jObij)OCjk = aiiO(bijOcjk) aiibifici,} = aijO(biiOcij) (20) aijbjjcjt = (aijObj) Ccjk = a1j O(bjOcjk) I atjbjicfk} = ai)O(bijOcik) aijbjkcki} = Ue.{ (aij3Objk)OCki} = UeiaijO(bjkOcki)} aijbjkCk 1} = (aijObjk)OCkI = aiJO(bJkOCk1) - Downloaded by guest on September 29, 2021 VOL. 56, 1966 MATHEMATICS: K. McCRIMMON 1075

We also have for distinct i,j Uaiiei = Uejaij2 (21) Vei = I, VU(ai)bi = VajbiVa On.flj. (22) 4. Standard Jordan Matrix Algebras.-If Z is an algebra with identity e and involution x -* x, then the operation X -X* = XI of taking the conjugate trans- pose defines the standard involution in the algebra Z,, of n xn matrices with entries in Z. Now suppose that Z is alternative (associative if n ) 4) and that Zo is a subspace of symmetric elements of the nucleus of Z containing the identity and all xyx for x E Z, y E Zo. The space t)(Ti,,Zo) of all *-symmetric elements of Z.n with diagonal entries in Zo is spanned by the elements x[ii] = xe1i (x C Zo) x[ij] = xet, + xeji (x C Z, i $ j) where the eij are the matrix units (e in the ij entry and zeros elsewhere). Note that our assumptions imply x + x E o for any x E Z, so if 1/2 E 4, then Zo = £t(Z) is the set of all symmetric elements of Z. If TZ = D(Z), then D(Zn,)o) is the set '(VZ) of all *-symmetric elements of Z.. The eii = e[ii] are n orthogonal idempotents with Peirce spaces tfi = Z~o[ii], 'N = Z [ij] (i $ j). In analogy with (18), (19), and (20) we introduce a quadratic representation U by UX[iiIy[ii] = (xyx) [ii] UI[iJ]y [ii] = (ayx) [ii] Ux[iJ]y[ij] = (xgx) [ij] x[ii]y[ij]z[jj]} = (xyz) [(i] x[ii]y[ij]z[ji]} = (xyz + xyz) [ii] (23) I x[ii]y[ij]z[jk]} = (xyz) [ik] { x[ii]y[ii]z[ij]} = (xyz) [0i] x[ij]y[jj]z[jk]} = (xyz)[ik] x[ij]y[ji]z[ik]} = (x(yz)) [ik] lx[ij]y[jk]z[kil] = (x(yz) + x(yz))[ii] Ix[ij]y[jk]z[kl]l = (xyz) [ii]. Note that parentheses are unnecessary in most products xyz because for x[ii] the element x E Zo is in the nucleus, for all x (E Z the element x + x E£ Zo is in the nucleus, and Z is alternative (or associative if there are four distinct indices). The assumptions on Zo guarantee that (Z,,Zo) is closed under this cubic composition. If Z is associative or 1/2 E ci, then (23) defines the usual quadratic representation UY = XYX or UxY = 2X- (X- Y) - X2- Y; in anycase the associated quadratic composition UxI = = XX is just the usual matrix composition. Since (4) and (5) are clearly satisfied, p(Zno) will be a Jordan algebra with identity I if and only if (2) and (3) hold, in which case it will be called a standard Jordan matrix algebra. We have seen that this is the case if Z is associative or £/2CE ; hence, in particular if n > 4. We conjecture that t(Z3,ZO) is always a Jordan algebra if Z is alternative and Zo a subspace of symmetric elements in the nucleus of Z containing the identity and all xyx for x C Z, y E Zo. To prove this directly seems a formidable task-even in the classical case the computations involved in proving the identity (1) of degree 4 are lengthy, and here we mupt prove Downloaded by guest on September 29, 2021 1076 MATHEMATICS: K. McCRIMMON PROC. N. A. S.

the identities (2) and (3) of degrees 7 and 5. However, for our purposes it suffices to consider only Z = ( a Cayley algebra with standard involution over a field 4) and Zo = 41 the set of all symmetric elements, and here we can use an elegant trick suggested by N. Jacobson. THEOREM OF THE EXCEPTIONAL ALGEBRA. If ( is a Cayley algebra with standard involution over a field 4), then the exceptional algebra 1D(S3) is a Jordan algebra with identity. Since 4) is a field, it suffices, by extending the base field, to prove the result for a split Cayley algebra. Such an algebra (S has a basis of eight elements

x = Ci, ej, eil, (eij)l (i = 1,2) where the involution is determined by 1= e2, j= -j, 1=-I (24) and the multiplication by e2 e, 2 1i2= 1ei +e2 (25) al = la a(bl) = (ba)l (al)b = (ab)l (al)(bl) = ba for a,b either ej or ej (i = 1,2). Then the algebra D(G) has a basis of 27 elements 1 [ii] (i = 1,2,3) and x [ij] (x as above, 1 < i < j s 3). Because the formulas (23), (24), and (25) are integral it suffices to prove the result for the integral subalgebra spanned by the 27 basis elements, and hence it is enough to consider the free split Cayley algebra over the integers. But this can be imbedded in the split Cayley algebras over the real field, and for the reals the result is known since (23) defines the usual quadratic representation. Thus even in characteristic 2 all of the basic Jordan algebras in the classical theory have analogues in the general theory. 5. Isotopes.-An element x of a Jordan algebra with identity is invertible with inverse y if Uxy = x, Uxy2 = 1. As usual, the following are equivalent: x is invertible, Ux is invertible, and 1 is in the range of Ux. In this case, the inverse y is uniquely determined as y = Uxulx, Ux and U, are inverses, and y is invertible with inverse x. We denote the inverse of x by x-l. Uzy is invertible if and only if x and y are, and then (Uzy)-l = U-'y-1; in particular, (x-1)2 = (x2)-l = x-2. If 21 c + is special, then y is the (Jordan) inverse of x in 21 if and only if y is the (associative) inverse of x in G. A Jordan algebra in which every nonzero element is invertible is called a Jordan division algebra. If u is invertible, the u-isotope of 21 = (1, U,1) is WU) (X,U(U),1(U)) where Ux(u) = UzUu, 1(u) = u-1 This is again a Jordan algebra with identity [that (5) holds in WM() equivalent to the isotope formula (2) in 2A], and we have the transitivity and symmetry relations {I1(u)} (v) = 21(UuV){I2(u)} (u-2) = 21. (26) Ideals or quadratic ideals coincide in 21 and its isotopes. We now consider certain isotopes of standard Jordan matrix algebras t(S"nz).

Let y = (a1,. .. ,-yn) be an n-tuple of invertible elements Pi SoT and or a permutation of {1,... ,n} of order 2 such that oyi = -i; -y is called a diagonal and 7r a Downloaded by guest on September 29, 2021 VOL. 56, 1966 MATHEMATICS: K. McCRIMMON 1077

twist. If G = diag {yi,. . . ,YnJ is the matrix with the es down its diagonal and P is the matrix of the permutation 7r (l's in the 7ri i places and O's elsewhere), then yTi= yi implies GP = PG. Since ri = yi and r2 = 1, we have G* = Gt = G, G-1 = diag {al-l,. 7,yn-1}, P* = P-1 = P so C = PG = GP is invertible in 1S(Z,,,Zo). Since the entries of C are in To, which is contained in the nucleus of X, we see C is in the nucleus of Z.. Thus X -> Xc = C-'XtC is an involution on Z" called a twisted diagonal involution (for y = 1 it is a twisted standard involution, for 7r = 1 a diagonal involution, and for -y = ir = 1 just the standard involution). The mapping X -- XC is a bijection of the isotope ID(ZnZo)(c) onto the set (Zn)nZoYtX r) (27)

of those c-symmetric elements X = Xc in 2n whose 7ri i entries X, ,i lie in Z0. We give this set the Jordan structure induced by the bijection and call it a twisted diagonal Jordan matrix algebra C(Tn, )O, Y, r) [or if -y = 1, a twisted standard Jordan matrix algebra '(ZnTO,7r); if or = 1, a diagonal Jordan matrix algebra .)(ZnTO,'Y); and if y = 7r = 1, just the standard Jordan matrix algebra D(Z~nO)]. In case Zo = St(Z) consists of all symmetric elements of X, we drop it from the notation. The Jordan structures of these algebras are determined by the isomorphisms

1t(ZmZWY,70T - t(n)ZO)(C '(Zn,_Z0,7) - 1t(ZnyiO)( ) ,tD(znS)OZ)-1 '!(Zn)ZO)( These structures are "natural" in that if Z is associative or 1/2 EC, then the quadratic representation is the usual one U.Y = XYX or UxY = 2X- (X. Y) - X2. Y and in any case UxI = X2 = XX. 6. Coordinatization Theorems.-Two orthogonal idempotents el,e2 are connected if there is an element u12 in the Peirce space 2112 [as in (18) ] invertible in the algebra 211(el + e2) = 241 + 2112 + 2122; if u122 = el + e2, they are strongly connected. Con- nectedness and strong connectedness are transitive relations. STRONG COORDINATIZATION THEOREM. If 2 is a Jordan algebra whose identity element is the sum 1 = Zei of n ) 3 orthogonal idempotents ej strongly connected to el by ul CE !li, then 21 is isomorphic to a standard Jordan matrix algebra 1(ZnZO) under an isomorphism sending es into eii and ulj into eli + eg1. If 1 = Zei where the el are connected to el by uli C 2hz, then there is an isotope 21 Wm(v), v = 2vii for vii = uii-l = UeiUij-2 C W1s, whose identity is a sum 1 = .Ze of orthogonal idempotents ei = uji = U6iui 2 strongly connected to il = el by i= u1i and such that the Peirce spaces !Aij = !Kij coincide in 21 and 21. By (25) 21 *() for v = v-2 £E 2ii a "diagonal" element. GENERAL COORDINATIZATION THEOREM. If 21 is a Jordan algebra whose identity element is a sum 1 = Zei of n > 3 orthogonal idempotents ej connected to e1 by uli E 21i, then 21 is isomorphic to a diagonal Jordan matrix algebra t?(ZnZoY) under an isomorphism sending ei into eii and u1i into yfel1 + Tyleg (where 71 = 1). The most important coordinate pairs (Z,T0) are determined by HERSTEIN-KLEINFELD-OSBORN THEOREM.2 If ! is a *-simple alternative ring with involution * and To is a subspace of symmetric elements of the nucleus whose nonzero elements are invertible and which contains the identity and all x*yx for x E X, y C Zo, then either Downloaded by guest on September 29, 2021 1078 MATHEMATICS: K. McCRIMMON PROC. N. A. S.

(I) Z is a direct sum AEDE* of anti-isomorphic associative division rings with ex- change involution and Zo = '(Z) = A(1E11*), (II) Z) is an associative division ring A with involution and Zo is a subspace of symmetric elements A0 containing 1 and all E* b for 6 £ A, Bo E A0, (III) Z is a split quaternion algebra e over itscenter Q withstandardinvolutionand So = "D(Z) = Q1, (IV) Z is a Cayley algebra ( over its center Q with standard involution and Zo = ID(Z) = U1. 7. Structure Theory.-An absolute zero divisor is an element z such that U, = 0. An idempotent e $ 0 is primitive if the Peirce space 2[l(e) contains no proper idempotents 7 e,O and is absolutely primitive if 211(e) is a Jordan division algebra. The basic theorem' on quadratic ideals carries over to the general setting with certain complications in characteristic 2. QUADRATIC IDEAL THEOREM. If e3 is a minimal quadratic ideal in a Jordan algebra 91, then either (I) Qt = 1)b for b an absolute zero divisor, (II) Z'= Ub1forall b $ O in e but UZQ3 = 2 = 0, (III) 3 = Ue1 is the Peirce quadratic ideal determined by an absolutely primitive idempotent e $ 0. In case II, either there is an idempotent e $ 0,1 in 9, or else 9 has characteristic 2 (291 = 0), an identity element 1, and elements b,d in minimal quadratic ideals 3 = U1,,1, Z = UdW with b2 = d2 = 0, bOd = 1, Ubd = b, Udb = d. In this case, the element c = b + d has c2 = 1, and in the isotope A = 1(") the elements b and d are supplementary orthogonal absolutely primitive idempotents strongly connected by the element 1 in the Peirce space i/2,(b) = 91/2(d), where 21A(b) = e and Al(d) = Z. LEMMA. If 91 is a Jordan algebra whose identity is the sum 1 = el + e2 of two absolutely primitive idempotents ej, then for a12 in the Peirce space 9112 (i) either a122 = 0 or a12 is invertible, (ii) if a122 = 0, then Ua.gJ11 = Ua12922 = 0. The set of absolute zero devisors is 8 = Z12 E 9112 1Z122 = Z120a12 = 0 for all a12 E 9112}. Either (1) el and e2 are connected, (2) 9112 = 0, or (3) W12 contains absolute zero divisors (3 $0 ). 9 is simple if and only if 3 = 0, 112 0. We next come to a recent result of Marshall Osborn.4 Recall the definitions (16) and (26). OSBORN's THEOREM. A simple Jordan algebra whose identity is the sum of two absolutely primitive idempotents is either (I) A2+for A an associative division algebra, (II) a diagonal Jordan matrix algebra &(IA2,A0,Y) where A is an associative division ring with involution, AO is a subspace of symmetric elements containing 1 and all 5*5O5 for a C A, Bo C Ao, and 'y = ("Y1,72) for yj $ 0 in Ao, (III) a split Jordan algebra 9(Q, 1) of a quadratic form Q with identity 1 over a field Downloaded by guest on September 29, 2021 VOL. 56, 1966 MATHEMATICS: K. McCRIMMON 1079 Finally, we come to the structure theory proper. A Jordan ring (i.e., an algebra over the ring of integers) is called nondegenerate if it has no proper absolute zero divisors (U2 = 0 ==) z = 0). A Jacobson ring is a Jordan ring such that (Ji) the descending chain condition holds for Peirce quadratic ideals, (J2) each nonzero Peirce quadratic ideal contains a minimal quadratic ideal. These rings play a role in the Jordan theory analogous to that played by the Artin- ian rings in the associative theory. FIRST STRUCTuRE THEOREM. A nondegenerate Jacobson ring has an identity and is a direct sum of ideals which are simple nondegenerate Jacobson rings. A simple nondegenerate Jacobson ring has an identity which is a sum of connected orthogonal primitive idempotents. By twisting this ring if necessary we obtain an isotope whose identity is the sum of n connected orthogonal absolutely primitive idempotents; n is called the capacity of the isotope. Applying Osborn's theorem, the coordinatization theorems, and the Herstein-Kleinfeld-Osborn theorem, we obtain the SECOND STRUCTURE THEOREM. A simple nondegenerate Jacobson ring is either (I) a Jordan division ring, (II) a Jordan algebra 5(Q,1) of a quadratic form over afield Q, or (III) a twisted diagonal Jordan matrix algebra of capacity n > 2 of the following types: (i) An+ for A an associative division ring (n > 2), (ii) g)(Any) for A an associative division ring with involution (n > 2), (iii) !(ZVnl) for e = &a a split quaternion algebra with standard involution over the field (n > 3), (iV) (93,'Y) for (Y a Cayley algebra with standard involution over a field 2 (n = 3), (V) '(GY),(12)) for S a Cayley algebra with standard involution over a field Q of characteristic 2 (n = 3), (vi) V(AnAo,'7r) for A an associative division ring with involution of characteristic 2, A0a subspace of symmetric elements containing 1 and all 8506 for 6 E A, 0 E& A0, oY = (Pyi,. .,y,Yn) for as # 0 in A0, and 7r a permutation of .... .,n} of order two such that y7,=yi(n ) 2). * This work was supported by an AFOSR postdoctoral fellowship and AFOSR grant no. 335-63. The author would like to thank Profs. Lowell J. Paige and Nathan Jacobson for their encourage- ment during the preparation of this paper. 1 Jacobson, N., "Structure theory for a class of Jordan algebras," these PROCEEDINGS, 55, 243- 251 (1966). 2Ibid., p. 249. 3 Ibid., p. 244. 4To appear in these PROCEEDINGS. Downloaded by guest on September 29, 2021