- Email: [email protected]
Remarks concerning sums of three squares and quaternion commutator identities
Remarks Concerning Sums of Three Squares and Quaternlon Commutator ldentltles Dennis R. Estes
University of Southern California Los Angeles, Califmia 90007 and Olga Taussky
Califnniu Institute of Technology Pa.wdenu, Califmia 91125
Submitted by G. P. Barker
ABSTRACT A survey of identities connected with sums of three squares, e-dimensional subalgebras of the quatemions, and additive commutators is given. The study of the last is linked to a theorem already observed by Gauss and re-proved here in various ways. A study of generalizations in recent years is added.
1.
SUMS OF SQUARES
IDENTITIES
The Euler identity expressing a product of sums of four squares as a sum of four squares is well known and may be viewed, following Hamilton, as just the multiplicative property of quatemion norms. Similar algebraic structures (the real, complex and Cayley numbers) provide multiplicative identities for sums of 1, 2, and 8 squares; but, as Hurwitz observed, there is no complete analogy in any other dimension (see [28]). Legendre (among others) characterized integers which are sums of three squares of integers, and is credited in Dickson’s History of the Theory of Numbers with the observation that the product of two such numbers (3 X21 -63, for example) is not necessarily the sum of three squares (Dickson’s source is p. ThSorie des Notnbres, but the conclusion does not seem lecture by van der Waerden, Hamilton’s Entdeckung der it is pointed out that this result shows there is no identity
LINEAR ALGEBRA AND ITS APPLICATIONS 3.5279-285 Q Elsevier North Holland, Inc., 1981 52Vanderbilt Ave., New York, NY 10017
198 in Legendre’s to be there). In a
Quaternionen [32] of the form
(1981)
00243795/81/010279
279 + 7$02.50
280
DENNIS R. ESTES AND OLGA TAUSSKY
where F, G, H axe rational functions of x, y, z, u, v, w, a result which fits in with more recently obtained insights. Nevertheless, sufficient algebraic structure is present in three dimensions to generate identities comparable to the four square identity. First there is the vector cross product u x v. Also, three dimensional vectors may be viewed as quaternions without scalar parts, termed pure quuternions, and as such inherit the quaternion multiplication (the product need not however be pure). Under this identification 2( aX b) corresponds to the quatemion commutator ab - ba; i.e., twice the vector cross product corresponds to the Lie product derived from quatemion multiplication. In this algebraic framework, the Lagrange identity
2
(a,bi-aibi)2
l
(1)
is also a simple consequence of the multiplicative property of quatemion norms (of course, this identity is self-evident and holds with any positive integer replacing the index 3). Alternatively, we may consider the vector equivalent of (1):
where (I )I denotes the length of a vector and 0 denotes inner product. Since IjaXbll=llall IIbIIsin0 and aob=llal( IIbIIcos8, 8 the angle between a and b, (1’) is just 1 = sin2 8 + cos2 8.
2.
TWO DIMENSIONAL QUATERNION
SUBALGEBRAS
Such algebras K containing 1 contain also the coefficient field F and a Thus K=F[[]. Each quatemion cx pure quatemion [= c,i+c2i+c,k#0. satisfies the quadratic equation cy2- t(a)a+N(a)=O where the trae, t(a), is the scalar part of a and the norm, N(a), is the sum of the squares of the coefficients of ar. Since p=
-A/(.$)=
-(cf+c;+c;),
(2)
the two dimensional subalgebras which contain 1 are the ring extensions F[t] where t#O, t2= -m, and m is the sum of three squares from F. Conversely, any such extension appears as a two dimensional subalgebra of
SUMS
OF THREE
the quaternion These
extensions
G
@F
SQUARES
algebra
281
(see Deuring,
fall into three
(the simple case);
W1-Fbl/~2F~~l nonzero
radical).
E F and 2m#O
in the case
Springer,
F [E]
is isomorphic
F[S]
copies of F in the case G
Algebren,
categories:
<
to a direct
(the semisimple
E F and 2m=
If, in the last two cases,
1935,
p. 135).
is a field in the
we set a’=
sum of two case);
0 (the case -m,
case and of a
UE F, then
([-a)(,$+~) =O. Since [ is pure and nonzero, the algebra cannot be a division algebra and is therefore the ring M,(F) of 2 X 2 matrices over F, or F has characteristic 2 (see e.g., [16, p. 58, Theorem 2.71). Each quatemion with real coefficients can be expressed in the form a,+~,~ with a,, a, real and 5 the pure part normalized so that t2= - 1 [33. Individually, real quaternions thus behave as complex numbers with pure quatemions as the analogue of purely imaginary complex numbers. It is not surprising therefore that the real quatemion algebra is algebraically closed; in fact, terminate
each real quatemion
polynomial
in a single noncommuting
with exactly one monomial of highest degree has a quatemion
inderoot
(i.e., polynomials of the type u,xu,x. . . u,_,xa,+Cmi(x) where the degree of each mj(x) is less than n) [5]. This however is not the case should there be multiple monomials of highest degree. For example, the commutator polynomial ax-xa cannot represent any quatemion without trace zero. We give now several proofs of the fact that each pure quatemion is an additive commutator.
3.
ADDITIVE
COMMUTATORS
We seek quatemion where [=c,i +c,j+c,k case of characteristic
3.1.
Direct Approach If x=x,+x,i+x,j+r,k,
OF QUATERNIONS
solutions to the commutator equation xy - yx=[, is a fixed pure quatemion. Since xy - yx=O in the
2, we assume below that 220.
y=yO+
yli+
yzj+ ysk, then xy-yz=2(x,y,
-x3y2)i+2(x3y1--x1y3)j+2(x1y2-xZyl)k. It is therefore sufficient to express the vector f (ci, c,, cs) as the vector of cofactors’ of a 2 X3 matrix x1
x2
x3
Yl
Y2
Y3 1 .
(
‘The term “i th cofactor in ao (n - 1) X TVmatrix” is used here and later for (- l)‘+’ the determinant of the submatrix obtained by omitting the ith column.
times
282
DENNIS R. ESTES AND OLGA TAUSSKY
Choose numbers xl, x2, xa with x,#O to satisfy the equation c,x,+c,x,+ ~,x,=O.Lety,bearbitrary,y~=(x,y~+~~~)/3~~,andy,=(x~y,-~c,)/x,. It is easily checked that these values satisfy the desired conditions. 3.2.
Geometric Approach We assume for convenience that the real numbers are the coefficient field. The Lie product i (ab - ba), as previously observed, coincides with the vector product a X b on three dimensional Euclidean space. Consequently, the condition that 5 is an additive commutator is just the condition that each three dimensional vector is a cross product of two vectors. But this is geometrically obvious; i.e., we may take a, b to be two linearly independent vectors suitably normalized and in the plane orthogonal to 5. 3.3.
Algebraic Approach Case of a division algebra: If [#O is pure, then F [ [] is a field having the F-automorphism induced by 5-_) -t. The Noether-Skolem theorem implies that this automorphism is the restriction of an inner automorphism z+crxa- ’ on the quaternion algebra. Thus (=a-“($ at)-(t at)a-’ is an additive commutator. Case of M,(F): In this case E, -C; are two 2 X 2 matrices having the same minimal polynomial x2 + iV(6) = 0 whenever .$# 0 is pure. Since such 2 X 2 matrices are similar, we solve a
A Matrix Approach The argument given in the second case of (3.3) shows that any 2 X2 matrix having trace zero is an additive commutator. Shoda [23] was the first to observe that any square matrix having trace zero and entries in a field of characteristic zero is an additive commutator. The assumption that the coefficient field have characteristic zero was later removed by Albert and Muckenhoupt [2], and the result was further extended by Brown [4], who proved its validity in any classical Lie algebra. Set
and let C=[U,V]=UV-W. assume that
Since [U,V]=[U-u,,Z,V-u,,Z],
we may
283
SUMS OF THREE SQUARES in which case
[U,V]=( Thus, the condition
that
UlV2
-u1u2
u3v2
-
each
v3”2
2 X2
matrix
of trace
zero is the additive
commutator of 2 X 2 matrices is equivalent to the property ternions are additive commutators (assuming 220).
4.
RELATED
that pure quar-
REMARKS
(i) The above arguments
may be adapted
to show that an element
of
(reduced) trace zero in a central simple four dimensional algebra is an additive commutator (such algebras are but generalized quatemion algebras
PI). (ii) Gauss any 3-tuple
observed in Article 279 of his Disquisitims
of integers
is the tuple of cofactors
Arithmetics
of a 2 X3
integral
that matrix
(consequently, a pure quatemion having even integers as coefficients is an additive commutator of two integral quatemions). Hermite [ll, p. 2641 extended this result to integral row vectors of any dimension n. Following Lissner [17], we call a commutative ring R an outer product ring (OP-ring) if for each positive integer n and each 1 Xn row vector v having entries in R there is an (n - 1) X n matrix over R whose (n - 1) X (n - 1) cofactors (ordered lexicographically) are the entries of v [this is easily shown to be equivalent to the property that each D in the exterior power A”-‘( R”) is decomposable; i.e., there are vi ,..., v,_~ER” such that u=u,Av,/\~~~ AU,,]. Thus, Hermite’s result is that the ring Z of rational integers is an OP-ring. Lissner extended Hermite’s result to Dedekind domains, and Towber [31] proved that polynomial rings over Dedekind domains were also OP-rings. Towber’s local study in [30] provided
the result that a Noetherian
local ring is an
OP-ring if and only if its maximal ideal can be generated by two elements, and Hinohara [12] extended the result to semilocal Noetherian rings. Finally, Estes and Matijevic proved that the reduced Noetherian OP-rings are those reduced Noetherian rings whose orientable modules are free [A”(M)-R with n>OesM=R”] and whose localizations at maximal ideals are OP-rings
PI. (iii) A vector VEAL, m< n, is a Plucker vector if it satisfies the quadratic Plucker conditions (see [20]). In particular, each v ER”-A”-‘(W) is a Plucker vector. A commutative ring having the property that each Plucker vector is decomposable is a Pliicker or Towber ring. Consequently, Plucker rings are OP-rings. H. J. S. Smith [24] appears to have been the first to show that Z is a Plucker ring, although Gauss was aware of the fact that
284
DENNIS
Plucker vectors in
A”(Z” )
Towber
Arithmeticue).
are decomposable
initiated
the abstract
R. ESTES AND OLGA TAUSSKY
(see Article 236, Disquisitiuws study of Plucker
rings in [3I].
Geramita and Lissner [20] provided the necessary conditions for a Noetherian ring to be Plucker, namely that R has global dimension at most 2 and R-orientable conditions
modules are free, and Estes and Matijevic [6] proved that these suffice. Finally,
Kleiner
[13] established
that any Krull OP-ring
was a Plucker ring, and Kotlarsky [14, 151 has considered classifying non-Noetherian Plucker rings.
the problem
of
(iv) We observed in Sec. 3.4 that each 2 X 2 matrix having trace zero and entries in a commutative ring R is an additive commutator of two 2 X2 matrices over R if and only if each o ER3 is the vector of cofactors of a 2 X 3 matrix over R. Lissner [I91 observed that these properties fail in polynomial rings in three or more variables. In fact, if these conditions are present in a Noetherian ring R, then R has Krull dimension at most 2 [30], and if in addition R is an integerally closed domain, then R is a Plucker ring [6]. (v) Except for the classical Lie algebras and central simple 4 dimensional algebras, there seems to be little information on the problem of classifying the elements
in an algebra which are additive commutators.
In the case of
nonassociative algebras, Gordon [lo] has shown that any element of trace zero in an octonion algebra of characteristic #2 (the Cayley algebra, for example) is an additive commutator. (vi) Recently
Taussky [25] studied det(AB-BA)
for A, BEM,(
F). If at
least one of A, B has distinct eigenvalues not in F, but in a proper extension E, then -det(AB-BA)=norrn.,,A, GEE. It seems of interest to see whether norm( ab- bu), a, b quatemions, can be somehow linked to this result. Putting m = ~~a~ and r= Za i bi in (l), Sec. 1, we have - norm( ab - ba) = r2 -mZ;bf. This is slightly reminiscent of the result. REFERENCES 1
A. A. Albert, Structure of Algebras, AMS, Providence, R.I., 1961.
2
A. A. Albert and B. Muckenhoupt, On matrices of trace zero, Michigan Math. J.
4:1-3 3 4
(1947).
L. Brand, The roots of a quatemion, Amer. Math. Monthly 49:519-520 14:763-767
5
S. Eilenberg and I. Niven, The “fundamental theorem of algebra” for quatem-
Math. Sot. 50:246-248
D. R. Estes and J. R. Matijevic, complete intersections, 1. Algeha
7
Sot.
(1963).
ions, Bull. Aw.
6
(1942).
G. Brown, On commutators in a simple Lie algebra, Proc. Amer. Math.
-,
(1944).
Matrix factorization, exterior powers, and
58: 117- 135 (1979).
Unique factorization of matrices and Towber rings, J. Algebru 59:387-
394 (1979).
SUMS OF THREE
Local-global
8
-,
9
appear. A. V. Geramita, Math.
10
criteria
Polynomial
24:866-872
S. Robert
285
SQUARES for outer
product
rings,
Canad.
rings with the outer product
J.
Math.,
to
property,
Cat&.
I.
(1972).
Gordon,
Associators
in simple algebras,
Pacific
J. Math.
51:131-141
(1974). 11
C. Hermite,
Extraits
de lettres
de M. Ch. Hermite
objets de la tb&orie des nombres, 12
Y. Hinohara,
On semilocal
13
G. B. Kleiner, Krull OP-rings
14
M. Kotlarsky,
On
Nauk, 33:219-220 15 16
J. Reine Angew.
OP-rings,
Pliicker
Math.
Proc. Amer.
are Pliicker, properties
A M. Jacobi
Math.
sur d&rents
Math. 40:261-315
(1850).
Sot. 32:16-20
(1972).
7:287-305
USSR-Zzu.
of non-Noetberian
rings,
(1973). Mat.
Uspehi
(1978).
On Pliicker properties of commutative rings, J. Algebra, to appear. -1 T. Y. Lam, The Algehruic Theory of Quudrutic Forms, Benjamin, Reading, Mass., 1973.
17
D. Lissner,
18 19
-,
Outer product
rings, Truns. Amer.
OP-rings
and Seshadri’s theorem,
Matrices
over polynomial
Math.
Sot.
116:526-535
J. Algebra 5:362-366
rings, Trans. Amer.
Math.
(1965).
(1967). ,Soc. 98:295-305
0.’ 20
D. Lissner and A. V. Geramita,
21
-, (1970).
Remarks
Towber
22
I. Niven, Equations
23
K. Shoda, Einige S%tze tiber Matrizen,
24
26
Amer.
Cunad.
Math.
15:13-40
J. Math.
(1970). 22:1109-1117
Monthly 48:654-661
Japan J. Math.
13:361-365
(1941). (1937).
linear equations, in CoZZecctedWorks,
1894. of rational 2 X 2 matrices,
Linear Algebra and
12:1-6 (1975). Pairs of sums of three squares of integers whose product
pF&,
28
Oxford,
0. Taussky, Additive commutators Awl.
27
in quatemions,
rings,
H. J. S. Smith, On systems of indeterminate Vol. 1, Clarendon,
25
rings, .Z. Algebra
on OP and Towber
in Generul Zneguulities 2 (E. Beckenbach,
has the same
Ed.), Birkh%user, Basel, 1980.
Results concerning composition of sums of three squares, Lineur -, Multilinear Algebra 8230-233 (1980). Sums of squares, Amer. Math. Monthly 77:805X%30 (1970). -,
and
29
History of sums of squares in algebra, in Proceedings of the Conference -> on History of Algebra, El Paso, Tex., 1975, Texas Tech. Univ., to appear.
30
J. Towber, 14: 194
Complete
31 10:299-309 32
Local
rings
with
the
outer
product
property,
ZZZirwis I.
Math.
197 (1970). reducibility
in exterior algebras over free modules, 1. Algebm
(1968).
B. L. Van der Waerden, and Ruprecht,
Gijttingen,
Receiwd 19 March
1980
HamiZtons Entdeckung 1973.
der Quutemionen,
Vanderhoeck
University of Southern California Los Angeles, Califmia 90007 and Olga Taussky
Califnniu Institute of Technology Pa.wdenu, Califmia 91125
Submitted by G. P. Barker
ABSTRACT A survey of identities connected with sums of three squares, e-dimensional subalgebras of the quatemions, and additive commutators is given. The study of the last is linked to a theorem already observed by Gauss and re-proved here in various ways. A study of generalizations in recent years is added.
1.
SUMS OF SQUARES
IDENTITIES
The Euler identity expressing a product of sums of four squares as a sum of four squares is well known and may be viewed, following Hamilton, as just the multiplicative property of quatemion norms. Similar algebraic structures (the real, complex and Cayley numbers) provide multiplicative identities for sums of 1, 2, and 8 squares; but, as Hurwitz observed, there is no complete analogy in any other dimension (see [28]). Legendre (among others) characterized integers which are sums of three squares of integers, and is credited in Dickson’s History of the Theory of Numbers with the observation that the product of two such numbers (3 X21 -63, for example) is not necessarily the sum of three squares (Dickson’s source is p. ThSorie des Notnbres, but the conclusion does not seem lecture by van der Waerden, Hamilton’s Entdeckung der it is pointed out that this result shows there is no identity
LINEAR ALGEBRA AND ITS APPLICATIONS 3.5279-285 Q Elsevier North Holland, Inc., 1981 52Vanderbilt Ave., New York, NY 10017
198 in Legendre’s to be there). In a
Quaternionen [32] of the form
(1981)
00243795/81/010279
279 + 7$02.50
280
DENNIS R. ESTES AND OLGA TAUSSKY
where F, G, H axe rational functions of x, y, z, u, v, w, a result which fits in with more recently obtained insights. Nevertheless, sufficient algebraic structure is present in three dimensions to generate identities comparable to the four square identity. First there is the vector cross product u x v. Also, three dimensional vectors may be viewed as quaternions without scalar parts, termed pure quuternions, and as such inherit the quaternion multiplication (the product need not however be pure). Under this identification 2( aX b) corresponds to the quatemion commutator ab - ba; i.e., twice the vector cross product corresponds to the Lie product derived from quatemion multiplication. In this algebraic framework, the Lagrange identity
2
(a,bi-aibi)2
l
(1)
is also a simple consequence of the multiplicative property of quatemion norms (of course, this identity is self-evident and holds with any positive integer replacing the index 3). Alternatively, we may consider the vector equivalent of (1):
where (I )I denotes the length of a vector and 0 denotes inner product. Since IjaXbll=llall IIbIIsin0 and aob=llal( IIbIIcos8, 8 the angle between a and b, (1’) is just 1 = sin2 8 + cos2 8.
2.
TWO DIMENSIONAL QUATERNION
SUBALGEBRAS
Such algebras K containing 1 contain also the coefficient field F and a Thus K=F[[]. Each quatemion cx pure quatemion [= c,i+c2i+c,k#0. satisfies the quadratic equation cy2- t(a)a+N(a)=O where the trae, t(a), is the scalar part of a and the norm, N(a), is the sum of the squares of the coefficients of ar. Since p=
-A/(.$)=
-(cf+c;+c;),
(2)
the two dimensional subalgebras which contain 1 are the ring extensions F[t] where t#O, t2= -m, and m is the sum of three squares from F. Conversely, any such extension appears as a two dimensional subalgebra of
SUMS
OF THREE
the quaternion These
extensions
G
@F
SQUARES
algebra
281
(see Deuring,
fall into three
(the simple case);
W1-Fbl/~2F~~l nonzero
radical).
E F and 2m#O
in the case
Springer,
F [E]
is isomorphic
F[S]
copies of F in the case G
Algebren,
categories:
<
to a direct
(the semisimple
E F and 2m=
If, in the last two cases,
1935,
p. 135).
is a field in the
we set a’=
sum of two case);
0 (the case -m,
case and of a
UE F, then
([-a)(,$+~) =O. Since [ is pure and nonzero, the algebra cannot be a division algebra and is therefore the ring M,(F) of 2 X 2 matrices over F, or F has characteristic 2 (see e.g., [16, p. 58, Theorem 2.71). Each quatemion with real coefficients can be expressed in the form a,+~,~ with a,, a, real and 5 the pure part normalized so that t2= - 1 [33. Individually, real quaternions thus behave as complex numbers with pure quatemions as the analogue of purely imaginary complex numbers. It is not surprising therefore that the real quatemion algebra is algebraically closed; in fact, terminate
each real quatemion
polynomial
in a single noncommuting
with exactly one monomial of highest degree has a quatemion
inderoot
(i.e., polynomials of the type u,xu,x. . . u,_,xa,+Cmi(x) where the degree of each mj(x) is less than n) [5]. This however is not the case should there be multiple monomials of highest degree. For example, the commutator polynomial ax-xa cannot represent any quatemion without trace zero. We give now several proofs of the fact that each pure quatemion is an additive commutator.
3.
ADDITIVE
COMMUTATORS
We seek quatemion where [=c,i +c,j+c,k case of characteristic
3.1.
Direct Approach If x=x,+x,i+x,j+r,k,
OF QUATERNIONS
solutions to the commutator equation xy - yx=[, is a fixed pure quatemion. Since xy - yx=O in the
2, we assume below that 220.
y=yO+
yli+
yzj+ ysk, then xy-yz=2(x,y,
-x3y2)i+2(x3y1--x1y3)j+2(x1y2-xZyl)k. It is therefore sufficient to express the vector f (ci, c,, cs) as the vector of cofactors’ of a 2 X3 matrix x1
x2
x3
Yl
Y2
Y3 1 .
(
‘The term “i th cofactor in ao (n - 1) X TVmatrix” is used here and later for (- l)‘+’ the determinant of the submatrix obtained by omitting the ith column.
times
282
DENNIS R. ESTES AND OLGA TAUSSKY
Choose numbers xl, x2, xa with x,#O to satisfy the equation c,x,+c,x,+ ~,x,=O.Lety,bearbitrary,y~=(x,y~+~~~)/3~~,andy,=(x~y,-~c,)/x,. It is easily checked that these values satisfy the desired conditions. 3.2.
Geometric Approach We assume for convenience that the real numbers are the coefficient field. The Lie product i (ab - ba), as previously observed, coincides with the vector product a X b on three dimensional Euclidean space. Consequently, the condition that 5 is an additive commutator is just the condition that each three dimensional vector is a cross product of two vectors. But this is geometrically obvious; i.e., we may take a, b to be two linearly independent vectors suitably normalized and in the plane orthogonal to 5. 3.3.
Algebraic Approach Case of a division algebra: If [#O is pure, then F [ [] is a field having the F-automorphism induced by 5-_) -t. The Noether-Skolem theorem implies that this automorphism is the restriction of an inner automorphism z+crxa- ’ on the quaternion algebra. Thus (=a-“($ at)-(t at)a-’ is an additive commutator. Case of M,(F): In this case E, -C; are two 2 X 2 matrices having the same minimal polynomial x2 + iV(6) = 0 whenever .$# 0 is pure. Since such 2 X 2 matrices are similar, we solve a
A Matrix Approach The argument given in the second case of (3.3) shows that any 2 X2 matrix having trace zero is an additive commutator. Shoda [23] was the first to observe that any square matrix having trace zero and entries in a field of characteristic zero is an additive commutator. The assumption that the coefficient field have characteristic zero was later removed by Albert and Muckenhoupt [2], and the result was further extended by Brown [4], who proved its validity in any classical Lie algebra. Set
and let C=[U,V]=UV-W. assume that
Since [U,V]=[U-u,,Z,V-u,,Z],
we may
283
SUMS OF THREE SQUARES in which case
[U,V]=( Thus, the condition
that
UlV2
-u1u2
u3v2
-
each
v3”2
2 X2
matrix
of trace
zero is the additive
commutator of 2 X 2 matrices is equivalent to the property ternions are additive commutators (assuming 220).
4.
RELATED
that pure quar-
REMARKS
(i) The above arguments
may be adapted
to show that an element
of
(reduced) trace zero in a central simple four dimensional algebra is an additive commutator (such algebras are but generalized quatemion algebras
PI). (ii) Gauss any 3-tuple
observed in Article 279 of his Disquisitims
of integers
is the tuple of cofactors
Arithmetics
of a 2 X3
integral
that matrix
(consequently, a pure quatemion having even integers as coefficients is an additive commutator of two integral quatemions). Hermite [ll, p. 2641 extended this result to integral row vectors of any dimension n. Following Lissner [17], we call a commutative ring R an outer product ring (OP-ring) if for each positive integer n and each 1 Xn row vector v having entries in R there is an (n - 1) X n matrix over R whose (n - 1) X (n - 1) cofactors (ordered lexicographically) are the entries of v [this is easily shown to be equivalent to the property that each D in the exterior power A”-‘( R”) is decomposable; i.e., there are vi ,..., v,_~ER” such that u=u,Av,/\~~~ AU,,]. Thus, Hermite’s result is that the ring Z of rational integers is an OP-ring. Lissner extended Hermite’s result to Dedekind domains, and Towber [31] proved that polynomial rings over Dedekind domains were also OP-rings. Towber’s local study in [30] provided
the result that a Noetherian
local ring is an
OP-ring if and only if its maximal ideal can be generated by two elements, and Hinohara [12] extended the result to semilocal Noetherian rings. Finally, Estes and Matijevic proved that the reduced Noetherian OP-rings are those reduced Noetherian rings whose orientable modules are free [A”(M)-R with n>OesM=R”] and whose localizations at maximal ideals are OP-rings
PI. (iii) A vector VEAL, m< n, is a Plucker vector if it satisfies the quadratic Plucker conditions (see [20]). In particular, each v ER”-A”-‘(W) is a Plucker vector. A commutative ring having the property that each Plucker vector is decomposable is a Pliicker or Towber ring. Consequently, Plucker rings are OP-rings. H. J. S. Smith [24] appears to have been the first to show that Z is a Plucker ring, although Gauss was aware of the fact that
284
DENNIS
Plucker vectors in
A”(Z” )
Towber
Arithmeticue).
are decomposable
initiated
the abstract
R. ESTES AND OLGA TAUSSKY
(see Article 236, Disquisitiuws study of Plucker
rings in [3I].
Geramita and Lissner [20] provided the necessary conditions for a Noetherian ring to be Plucker, namely that R has global dimension at most 2 and R-orientable conditions
modules are free, and Estes and Matijevic [6] proved that these suffice. Finally,
Kleiner
[13] established
that any Krull OP-ring
was a Plucker ring, and Kotlarsky [14, 151 has considered classifying non-Noetherian Plucker rings.
the problem
of
(iv) We observed in Sec. 3.4 that each 2 X 2 matrix having trace zero and entries in a commutative ring R is an additive commutator of two 2 X2 matrices over R if and only if each o ER3 is the vector of cofactors of a 2 X 3 matrix over R. Lissner [I91 observed that these properties fail in polynomial rings in three or more variables. In fact, if these conditions are present in a Noetherian ring R, then R has Krull dimension at most 2 [30], and if in addition R is an integerally closed domain, then R is a Plucker ring [6]. (v) Except for the classical Lie algebras and central simple 4 dimensional algebras, there seems to be little information on the problem of classifying the elements
in an algebra which are additive commutators.
In the case of
nonassociative algebras, Gordon [lo] has shown that any element of trace zero in an octonion algebra of characteristic #2 (the Cayley algebra, for example) is an additive commutator. (vi) Recently
Taussky [25] studied det(AB-BA)
for A, BEM,(
F). If at
least one of A, B has distinct eigenvalues not in F, but in a proper extension E, then -det(AB-BA)=norrn.,,A, GEE. It seems of interest to see whether norm( ab- bu), a, b quatemions, can be somehow linked to this result. Putting m = ~~a~ and r= Za i bi in (l), Sec. 1, we have - norm( ab - ba) = r2 -mZ;bf. This is slightly reminiscent of the result. REFERENCES 1
A. A. Albert, Structure of Algebras, AMS, Providence, R.I., 1961.
2
A. A. Albert and B. Muckenhoupt, On matrices of trace zero, Michigan Math. J.
4:1-3 3 4
(1947).
L. Brand, The roots of a quatemion, Amer. Math. Monthly 49:519-520 14:763-767
5
S. Eilenberg and I. Niven, The “fundamental theorem of algebra” for quatem-
Math. Sot. 50:246-248
D. R. Estes and J. R. Matijevic, complete intersections, 1. Algeha
7
Sot.
(1963).
ions, Bull. Aw.
6
(1942).
G. Brown, On commutators in a simple Lie algebra, Proc. Amer. Math.
-,
(1944).
Matrix factorization, exterior powers, and
58: 117- 135 (1979).
Unique factorization of matrices and Towber rings, J. Algebru 59:387-
394 (1979).
SUMS OF THREE
Local-global
8
-,
9
appear. A. V. Geramita, Math.
10
criteria
Polynomial
24:866-872
S. Robert
285
SQUARES for outer
product
rings,
Canad.
rings with the outer product
J.
Math.,
to
property,
Cat&.
I.
(1972).
Gordon,
Associators
in simple algebras,
Pacific
J. Math.
51:131-141
(1974). 11
C. Hermite,
Extraits
de lettres
de M. Ch. Hermite
objets de la tb&orie des nombres, 12
Y. Hinohara,
On semilocal
13
G. B. Kleiner, Krull OP-rings
14
M. Kotlarsky,
On
Nauk, 33:219-220 15 16
J. Reine Angew.
OP-rings,
Pliicker
Math.
Proc. Amer.
are Pliicker, properties
A M. Jacobi
Math.
sur d&rents
Math. 40:261-315
(1850).
Sot. 32:16-20
(1972).
7:287-305
USSR-Zzu.
of non-Noetberian
rings,
(1973). Mat.
Uspehi
(1978).
On Pliicker properties of commutative rings, J. Algebra, to appear. -1 T. Y. Lam, The Algehruic Theory of Quudrutic Forms, Benjamin, Reading, Mass., 1973.
17
D. Lissner,
18 19
-,
Outer product
rings, Truns. Amer.
OP-rings
and Seshadri’s theorem,
Matrices
over polynomial
Math.
Sot.
116:526-535
J. Algebra 5:362-366
rings, Trans. Amer.
Math.
(1965).
(1967). ,Soc. 98:295-305
0.’ 20
D. Lissner and A. V. Geramita,
21
-, (1970).
Remarks
Towber
22
I. Niven, Equations
23
K. Shoda, Einige S%tze tiber Matrizen,
24
26
Amer.
Cunad.
Math.
15:13-40
J. Math.
(1970). 22:1109-1117
Monthly 48:654-661
Japan J. Math.
13:361-365
(1941). (1937).
linear equations, in CoZZecctedWorks,
1894. of rational 2 X 2 matrices,
Linear Algebra and
12:1-6 (1975). Pairs of sums of three squares of integers whose product
pF&,
28
Oxford,
0. Taussky, Additive commutators Awl.
27
in quatemions,
rings,
H. J. S. Smith, On systems of indeterminate Vol. 1, Clarendon,
25
rings, .Z. Algebra
on OP and Towber
in Generul Zneguulities 2 (E. Beckenbach,
has the same
Ed.), Birkh%user, Basel, 1980.
Results concerning composition of sums of three squares, Lineur -, Multilinear Algebra 8230-233 (1980). Sums of squares, Amer. Math. Monthly 77:805X%30 (1970). -,
and
29
History of sums of squares in algebra, in Proceedings of the Conference -> on History of Algebra, El Paso, Tex., 1975, Texas Tech. Univ., to appear.
30
J. Towber, 14: 194
Complete
31 10:299-309 32
Local
rings
with
the
outer
product
property,
ZZZirwis I.
Math.
197 (1970). reducibility
in exterior algebras over free modules, 1. Algebm
(1968).
B. L. Van der Waerden, and Ruprecht,
Gijttingen,
Receiwd 19 March
1980
HamiZtons Entdeckung 1973.
der Quutemionen,
Vanderhoeck