AN IDENTITY IN JORDAN RINGS
MARSHALL HALL, JR.
1. Introduction. An abstract Jordan ring / is a distributive
which multiplication
satisfies the two laws
(1.1)
ba = ab,
(1.2)
ia2b)a = a2iba).
We shall assume here that division by two is
cisely that x+x = b has a unique solution x for
of course be true if / is an algebra over a
different from 2. Jacobson [S] has defined the
ring in
possible, or more preany given b. This will
field of characteristic
Jordan triple product
{abc} by the rule
(1.3)
,
.
1
1
[abc] = —iab)c -\-{bc)a-iac)b.
1
A special Jordan ring 5 may be formed from an associative ring R,
by taking the elements of R with the same addition but with a
Jordan product ab given in terms of the associative product a-b of R
by the rule
(1.4)
ab = a-b + ba.
From (1.4) the laws (1.1) and (1.2) follow and the triple product
simplifies to the form
(1.5)
{abc} = a-b-c + c-b-a.
In this paper the identity
(1.6)
{aba}2 = {a{ba2b}a}
is proved to hold in abstract Jordan rings. This is immediate for special Jordan rings, but was posed as an unsolved problem by Jacobson
in his Colloquium lectures at Ann Arbor in September,
1955. In this
paper the identity is proved by finding a partial basis for the free
Jordan ring with two generators,
the basis being found for all elements of degree at most 5 and for elements of degree 4 in a and degree
2 in b. Expressing both sides of (1.6) in terms of the basis, the two
sides are found to be identical.
Using this identity, Jacobson [5] has been able to prove certain
structure and representation
theorems for abstract Jordan algebras
which had been proved previously by Albert [l] for special algebras.
Received by the editors January 24, 1956.
990
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
991
AN IDENTITY IN JORDAN RINGS
A further interesting by-product of this paper is that, so far as it
goes, the basis indicates that the free Jordan ring is special. This
comes from a comparison with results established by P. M. Cohn [4].
Another proof of the identity has been found independently
by
Mr. Lawrence Harper of the University of Chicago. He does not find
a basis, but uses a sequence of relations which are consequences of
(1.1) and (1.2).
2. The monomial relations of a Jordan ring. In a Jordan ring addition is an abelian group, both distributive
plication must satisfy the two laws
(2.1)
xy = yx,
(2.2)
(y2x)y = y2(xy).
laws hold and the multi-
We assume also that division by 2 is possible, or more precisely that
the additive group contains no element of order 2. This will of course
be true if the ring admits as operators a field of characteristic
different
from 2.
In a ring generated
expressed as a linear
tions, it is sufficient
trivial to verify that
ments if it holds for
In (2.2) replace y
by elements ax, • • ■ , an every element can be
combination of monomials, and to find all relato find the relations on the monomials. It is
the commutative law (2.1) will hold for all eleall monomials.
by y+z and from the result subtract both (2.2)
and (2.2) with y replaced by z. This gives
(y2x)z + 2((yz)x)y
+ 2((yz)x)z + (z2x)y
= y2(xz) + 2(yz)(xy) + 2(yz)(xz) + z\xy).
In (2.3) replace z by z+w and from this subtract
both (2.3) and (2.3)
with z replaced by w. Dividing by 2 we get
/„ „n
(xiyz))w+
(2.4)
(x(yw))z+
(x(zw))y
= ixw)iyz) + ixz)iyw)
+ ixy)izw).
With w = y and it = zwe may derive (2.3) from (2.4), while if w = z = y
(2.4) becomes three times (2.2). Using induction on the number of
monomials involved, it is easy to show that (2.1) and (2.2) hold for
all elements if and only if (2.1), (2.2) and (2.4) hold for all monomials.
We note that
(2.4) is symmetric
in y, z, w.
3. Basis elements in the free Jordan ring with two generators. We
shall find a
Jordan ring
(2.4) do not
for the first
basis for elements of degrees one through five in the free
J generated by two elements a and b. Since (2.2) and
apply below the fourth degree, we can write down a basis
three degrees immediately.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
992
MARSHALLHALL,JR.
(3.1)
[December
Degree 1
a,
b,
Degree 2
a2, ab, b2,
Degree 3
a8, a2b, iab)a, iab)b, ab2, 6*.
Here we have written a2a = a* and b2b = b*. It is known that a Jordan
ring is power associative. If we write a1 = a, and recursively an = an~1a,
then it will be true that if t+u = n then a'aM = an, and the ring generated by a is associative as well as commutative.
From (2.2) with
x = an~3, y = awe get (an-3a2)<z = (an-2)a2 and so by induction
ian~2)a2
= an. If in (2.4) we put x = a', y = ar, z = w = a, we obtain 2(ar+,+1)a
+ (a*+2)ar = 2(a*+1)ar+, + (a'+r)a2. By taking appropriate
values for j
and r we may complete
the relations
a" = an_1a = an~2a2 = an~3a3
= ■ ■ ■ =atau, naturally using induction on n, and one of 5 or r.
For degree 4 with each of x and y taking the values a and b in (2.2)
we get the four relations
ia2a)a = cV
(3.2)
ib2b)b = b2b2 = bl,
ib2a)b = b2iab),
From (2.4) we obtain further
(3.3)
= a4,
ia2b)a = a2iba).
relations
of degree four.
x
y
z
w
a
a
b
b
2(o(a6))6 + iab2)a = 2(a¿)2 + a2b2,
a
b
a
a
2(a(a6))a +
a*b
= 3a2(a6),
b a
b b
2ibiab))b +
b3a
= 3b2iab),
b
a
2ibiab))a + ia2b)b = 2(a&)2 + a2J2.
b
a
From (3.2) and (3.3) we can now give a basis for degree 4 and express
the remaining monomials in terms of this basis.
Degree 4 basis and relations.
a4; ia2b)a, Hab)a)a; Hab)b)a, Hab)a)b, iab)2, a2b2;
iab2)b, Hab)b)b; 64.
a\ab) = ia2b)a, b\ab) = iab2)b,
(3.4)
a3&= 3ia2b)a - 2((oi)a)a,
ab* = 3iab2)b - 2((oi)6)ft,
(ai>2)a*= a2b2 + 2(ö&)2 -
2((aft)a)ft,
(a2b)b = a-b2 + 2(a6)2 -
2((oi)i)o.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1956]
AN IDENTITY IN JORDAN RINGS
In finding a basis for degree 5, it is convenient
993
to label the mono-
mials:
(3.5)
ux = iiiab)a)a)a,
vx = (((ab)b)a)a,
u2 = ((a2b)a)a,
v2 = (((ab)a)b)a,
M3 = a4b,
»3 = (a2b2)a,
«4 = ((ab)a)a2,
»4 = (ab)2a,
iii = (a2b)a2,
»6 = (((ab)a)a)b,
M6 = a\ab),
»6 = ((a2b)a)b,
u-i = (a2(ab))a
= u2,
v¡ = (ab2)a2,
«s = (a*b)a = — 2«i + 3m2,
»s = ((ab)b)a2,
»9 = ((ab)a)(ab),
»10 = (a2b)(ab),
»11 = a362,
»i4 = ((aô2)a)a
= — 2»2 -f »3 + 2»4,
flu = ((a2b)b)a
= — 2»i + »3 + 2»4,
»h = (a2(ab))b
= »6,
»15 = (a*b)b = — 2»s + 3»6.
The relations here are derived from (3.5) as may be easily verified.
There will also be the monomials obtained by interchanging
a and b
in the above list, as well as the monomials ab and b5. Substituting
in
(2.2) we find
x = ab, y = a,
ut = w7,
x = b2, y = a,
»7 = »3.
(3.6)
The law (2.4) gives the following
(3.7)
relations
on the w's:
x
y
z
w
a2
a
a
b
u-¡ + 2u-¡ = «5 + 2uo,
a
a2
a
b
u2 + u% + «4 = «s + 2m6,
a
ab
a
a
w6 + 2«i = 3w4,
b
a2
a
a
u$ + 2m8 = w6 + 2u%.
We also find from (2.4) the following relations
on the »'s.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
994
(3.8)
MARSHALLHAIL, JR.
[December
x
y
z
W
a2
b
b
a
»3 + 2»i4 = »11 + 2»io,
ab
a
a
b
»14 + 2»4 = »8 + 2»9,
a
a2
b
b
2»6 + »7 = 2»io + »a,
a
ab
a
b
»6 + »9 + vx = »8 + 2»9,
a
b2
a
a
»a + 2»i2 = 3»7,
b
a2
a
b
»16 +
b
ab
a
a
2»2 + »io = »s + 2»B.
»13 +
»8 = 2»io +
From these we may find a basis and relations
Type
a6 :
Type a*b:
»11,
for degree 5.
a6 = a4 a — a3a2.
Basis ux, u2, «3.
U\ =
«2,
«6 = 4«i — Au2 + Us,
«s = — 2«i + 3u2,
«7 = u2,
m8 = — 2«i + 3u2.
(3.9)
Type as62:
Basis »i, »2, »3, »4, »6, »e»7
=
»3,
»8 = 2»i — 2»4 + 2»6 — »6,
»9 =
— »1 +
»10 =
— 2»2 +
2»4 — »6 +
2»4 +
»«,
»6,
»11 = 4»2 + »3 — 4»4,
»12 =
— 2»2 + »3 +
2»4,
»13 =
— 2»i + »3 +
2»4,
»14 =
»6,
»18 =
— 2»6 + 3»6.
By interchanging
a and b throughout,
we will also find a basis of 6
elements of type a2bz, 3 elements of type a&4and the single element b6.
Thus there are a total of 20 basis elements of degree five.
4. Basis and relations for terms of type a*b2.Since the identity depends on terms which are all of type a4£>2,it will be sufficient to find a
basis and relations for the monomials of this type. Using the relations
of (3.4) and (3.9) any element of type a*b2 can be expressed in terms
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
19561
AN IDENTITYIN JORDANRINGS
995
of the following twenty one monomials:
xi = (i((àb)b)a)a)a
(4.1)
= via,
xn = (((ab)a)b)a2,
Xi = ((((ab) a) b) a) a = v2a,
xi2 = (ab)2a2,
x3 = ((a2b2)a)a
= v3a,
xn = (a2b2)a2,
Xi ■« ((ab)2a)a
= z)4a,
xu = (((ab)a)a)(ab),
xt — ((((ab)a)a)b)a
= v^a,
xu = ((a2b)a)(ab),
xo = (((a2b)a)b)a
= vsa,
xie = a*b2,
xi - ((((ab)a)a)a)b
= uib,
x17 = a3(a62),
x8 = (((a2b)a)a)b
= m26,
Xi8 = as((a¿>)¿>),
x9 = (a*b)b
= «36,
Xi9 = ((ab)a)2,
xu = (((ab)b)a)a2,
x20 = (a2b)((ab)a),
xn = (a2b)2.
We will also label the remaining
terms of the above list.
monomials
and express them in
x22 = ((ab2) a2) a
= v-¡a = x3,
x23 = (((ab)b)a2)a
= vga = 2xi — 2x4 + 2x6 — x»,
xu = (((ab) a) (ab)) a = vsa = — Xi + 2x4 — x&+ x6,
(4.2)
#26 = ((a2b)(ab))a
= Viaa = — 2x2 + 2x4 + x6,
xu = (<z362)a
= aua = 4x2 + x3 -\-4x4,
x27 = (((ab2) a) a) a
= vna = — 2x% + x3+
X28 = (((a2b)b)a)a
= vi3a = — 2xi + x%+ 2x4,
X29 = (((ab)a2)b)a
= vua = x6,
X30 = ((a3b)b)a
= vna = — 2x6 + 3x6,
X31 = (((a£>)a)a2)Z>
= w4£> = x8,
#32 = ((a2b)a2)b
= u6b = 4x7 — 4x8 + x9,
X33= (a3(ab))b
= u*b = — 2x7 + 3xg,
x34 = ((a2(ab))a)b
= mb = x8,
#35 = ((a%b)a)b
= u3b = — 2x7 + 3x8,
X36 = ((ab2)a)a2
= xi8 + 2xi2 — 2xn,
X37 = ((a2b)b)a2
= xu + 2xi2 — 2xi0,
x38 = (a2(ab))(ab)
= xn,
x8» = (a*b)(ab)
= 3xi6 — 2xi4.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
2x4,
996
MARSHALLHALL, JR.
The law (2.2) has only two applications
to elements
x = ab2,
y = a,
x22 = x36,
x = (ab)b,
y = a,
X23 = xi0.
(4.3)
There are, however,
x
[December
twenty-two
applicable
of type a4ft2
cases of (2.4):
y
z
w
1. a5,
a,
b,
b,
2x33 + x26 = *ie + 2*»,
2.
b,
a,
a,
2x26 + X32 = 2xu + X37,
3. (ab)a, b,
a,
a,
2x24 + x3i = 2xu + xxx,
4. a2,
a2,
b,
b,
2xi2 + x]3 = 2x2i + xu,
5.
a2,
ab,
a,
b,
x%x+ x23 + x38 = x20 + xi8 + x3s,
6. a2,
b2,
a,
a,
2x2I + xu = in + 2xi7,
7.
ab,
a2,
a,
b,
x33 + x26 + xx2 = x38 + x2o + xx»
8. ab,
ab,
a,
a,
2x2i + x38 = xx2 + 2xi9,
9. b2,
a2,
a,
a,
2xn + *m = 1*a + xu,
a8,
ft,
J,
2x36 + *« = *ie + 2x39,
(4.4) 11. a,
a2b, a,
b,
*g + Xu + Xjo — 2*« + «»7,
12. a,
(aft)a, a,
ft,
x7 + x2 + Xi9 = 2xu + *ii,
a,
a,
2x27 + xx-¡ = 3x%%,
(aft)ft, a,
a,
2xx + xx» = 3xio,
a2b,
10. a,
13. a,
14. a,
ab2,
15. ft,
c8,
a,
ft,
x9 + x30 + xis = 2x39 + x«,
16. ft,
a2ft,
a,
a,
2xo + x2X = 2xi6 + x37,
(ab)a, a,
a,
2xb + x2o = 2xi4 + xn,
17. ft,
18.
a,
a2,
aft,
ft,
x34 + xu + xi0 = xi8 + x20 + x38,
19.
a,
a2,
ft2,
a,
x3 + x16 + x36 = 2xi7 + xn,
20. a,
aft,
aft,
a,
2xXi + X4 = 2xi9 + X12,
21.
ft,
a2,
a2,
ft,
2x37 + x9 = Xi6 + 2x21,
22.
ft,
a2,
aft,
a,
x29 + x39 + Xn = xî0 + xxs + x3S.
From these 24 relations we find a basis and express the remaining
monomials in terms of this basis.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1956]
AN IDENTITY IN JORDANRINGS
997
Basis: *i, x¡, x%,x,, xh, x6, *7, x%,*n
(4.5)
Xi =
— 4xi + 4x2 + *i +
*io =
x¡i =
2*i
— 2*i
2xt —
x¡
— 4*7 + 6*»,
— 2*4 + 2xt — *»,
+ 4*4 — 2*j + 2*e
+
xt — 2*14,
Xn =
4*i + 2*2
— 7*4 -f- 4*t — 4*« + 2*7 — 2*s + 2*h,
*ia = — 12*i — 4*2 + *j + 22*4 — 12*6 + 12*« — 4*7 + 6*8 — 8*n,
*i6 =
2*i
— 3*4 + 2*s — 2*«
+2*h,
*16 = — 12*1 + 4*2 + *» + 14*4 — 12*6 + 12*( — 4*7 + 6*8 — 8*14,
*17 =
+ 4*2 + *3 — 4*4,
*is =
4*1
— 6*4 +
6*« — 3*«,
*U =
— 2*1 — *2
+
4*4 — 2*6 +
2*J — Xl +
*8,
*20 =
— 2*1
+
4*4 — 4*6 +
2*«
Xi,
*21 =
— 4*1
+ *8 +
The remaining
monomials
+
6*4 — 4*6
+ 2*g.
may of course be expressed in terms of
these by using (4.2).
5. Proof of the identity. In a Jordan
ternary operation abc by the rule
(5.1)
ring, Jacobson
{abc} = — (ab) c + —(be)a-(ac)
The identity
defines a
b.
which we wish to prove is the following:
(5.2)
{aba}2 = {a{ba2b}a}.
This is known to hold in special Jordan rings, but here we show that
it is true in every Jordan ring. Using (5.1) we express both sides of
(5.2) in terms of monomials.
{aba}2 = ((ab)a)2 -
(5.3)
(a2b)((ab)a)
1
4
+ — (a2b)2
1
=
X10 — x20 H—— X21,
4
{a{ba2b}a} = (((a2b)b)a)a-((a2b2)a)a-((a2b)b)a-
(5.4)
1
4
H-(a2b2)a2
111
2
= x2i-x3-x37
2
H-X13.
4
This identity is indeed true since using the relations of (4.2) and (4.4)
both expressions reduce to
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
998
(5.5)
MARSHALL
HALL,JR.
-xi
- x, + — x3 + — x4 + xt - x7 H-x».
4
2
2
6. Concluding remarks.1 Comparison of the results found here with
those of P. M. Cohn [2] is of some interest. In §3 we found all relations on the free ring / with two generators for elements of degrees
not exceeding five, and from these found the number of basis elements
for degrees one through five to be respectively 2, 3, 6, 10, 20. Cohn has
shown that the special free ring with two generators has a basis of the
reversible elements. This basis will consist of monomials which are
their own reverses together with elements which are the sum of a
monomial and its reverse, the reverse being distinct. This leads to a
basis with 2n~1+2r elements of degree n when n = 2r + \ or 2r+2.
Thus from a consideration of the number of basis elements alone, it
follows that J modulo the ideal Kt oí elements of degree six or higher
is a special ring, agreeing with the free special ring modulo elements
of degree six or higher. In particular, every identity on two generators of degree not exceeding five which holds in a special Jordan
ring also holds in any Jordan ring with two generators. Similar considerations apply if we include also the elements of degree four in one
variable and two in the other since there will be nine reversible basis
elements of this type in the free special Jordan ring. To prove that
the free Jordan ring with two generators is special it would be sufficient to show that for n = 2r+\ or 2r + 2 there are at most 2B-1 + 2r
basis elements of degree n.
Bibliography
1. A. A. Albert, On Jordan algebras of linear transformations,
Trans. Amer. Math-
Soc, vol. 59 (1946) pp. 524-555.
2. -,
A structure theory for Jordan algebras, Ann. of Math. vol. 48 (1947) pp.
546-567.
3. -,
A theory of power-associative commutative algebras, Trans. Amer. Math.
Soc.vol. 69 (1950)pp. 503-527.
4. P. M. Cohn, On homomorphic images of special Jordan
algebras, Canadian
Journal of Mathematics vol. 6 (1954) pp. 253-264.
5. N. Jacobson A theorem on the structure of Jordan algebras, To appear in Proc.
Nat. Acad. Sei. U.S.A.
Ohio State University
1 Added in proof: A Jordan ring over a field of characteristic
different from two is
special. See A. I. Shirshov, On special J-rings, Rec. Math. (Mat. Sbornik) N.S. vol. 38
(80) (1956) pp. 149-166.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use