A. A. Albert

Structure of algebras

Fundamental concepts Ideals and nilpotent algebras The structure theorems of Wedderburn Simple algebras Crossed products and exponents Cyclic semi-fields Cyclic algebras and

Power-associative rings

p

Symmetric and alternate matrices in an arbitrary field. I

algebras Representations and Riemann matrices Rational division algebras Involutions of algebras Special results.

On Reduced Exceptional Simple Jordan Algebras

We use the term ring for any additive abelian group closed with respect to a product operation such that the two-sided distributive law holds. When the associative law for products also holds we call the ring an associative ring. Every element x of any ring 21 generates a subring 2I(x) of 21 consisting of all finite sums of terms each of which is a finite product whose factors are all equal to x. We call 21 a power-associative ring when every 2I(x) is an associative subring of 21. We have shown elsewhere(*) that a ring 21 whose characteristic is zero is power-associative if and only if xx2=x2x and x2x2 = (x2x)x for every x of 21. This result is also true for all commutative rings having characteristic prime to 30, and the stated restrictions on the characteristic are actually necessary. Our present investigation begins with a derivation of results on the decompositions of a power-associative ring relative to its idempotents. When e is an idempotent of a commutative power-associative ring 21, the corresponding (right) multiplication Re is an endomorphism of A having simple elementary divisors and roots 0, 1/2, 1. There is a resulting decomposition of 21 as the supplementary sum 2íe(l)+2le(l/2)-|-2Ic(0) of submodules 2íe(X) such that xe=Xx. Moreover the multiplication relations for these submodules are nearly those holding for the case(2) where 21 is a Jordan ring. However, the situation becomes much more complicated when 21 is not commutative since then the elementary divisors of 7^e need not be simple and the characteristic roots are quite arbitrary. It is true, nevertheless, that a decomposition theory may be obtained for all power-associative rings 21 in which the equation 2x = a has a unique solution x in 21 for every a of 21. In this case we may always attach to 21 a commutative ring 2í(+) which is the same additive group as 21 and which has a product x-y defined in terms of the product xy of 21 by 2(x-y) =xy+yx. The ring 2I<+) is power-associative when 21 is, and every idempotent of 21 is also an idempotent of 2I(+). This yields a decomposition of 2I = 2Ie(l)-r-2I(!(l/2) + 2Ie(0) where 2le(X) is the set of all x such that xe+ex = 2Xx, and the submodules always have some of the multiplicative properties of the Jordan

On the Construction of Riemann Matrices II

The elementary theorems of the classical treatment of symmetric and alternate matrices may be shown, without change in the proofs, to hold for matrices whose elements are in any field of characteristic not two. The proofs fail in the characteristic two case and the results cannot hold since here the concepts of symmetric and alternate matrices coincide. But it is possible to obtain a unified treatment. We shall provide this here by adding a condition to the definition of alternate matrices which is redundant except for fields of characteristic two. The proofs of the classical results will then be completed by the addition of two necessary new arguments. The theorems on the definiteness of real symmetric matrices have had no analogues for general fields. They have been based on the property that the sum of any two non-negative real numbers is non-negative. This is equivalent to the property that for every real a and b we have a 2+b2 =c2 for a real c. But a2?b2 = (a+b)2 in any field of characteristic two and we shall use this fact to obtain complete analogues for arbitrary fields of characteristic two of the usual theorems on the definiteness of real symmetric matrices. Quadratics forms may be associated with symmetric matrices and the problem of their equivalence is equivalent to the problem of the congruence of the corresponding matrices. This is true except when the field of reference has characteristic two where no matric treatment has been given. We shall associate quadratic forms in this case with a certain type of non-symmetric matrix and shall use our results on the congruence of alternate matrices to obtain a matrix treatment of the quadratic form problem. The classical theoremst on pairs of symmetric or alternate matrices with complex elements will be shown here to be true for matrices with elements in any algebraically closed field whose characteristic is not two. This will be seen to imply that any two symmetric (or alternate) matrices are orthogonally equivalent if and only if they are similar. But the proof fails for fields of characteristic two.

A quadratic form problem in the calculus of variations

In this paper we shall be concerned only with finite dimensional algebras over an arbitrary field \( \mathfrak{F} \) of characteristic not two. Let \( \mathfrak{A} \) be an associative algebra over \( \mathfrak{F} \) and ab the associative product composition of \( \mathfrak{A} \). Then the vector space \( \mathfrak{A} \) is a Jordan algebra \( {\mathfrak{A}^{\left( + \right)}} \) relative to the composition a·b = 1/2(ab + ba), that is, this composition satisfies the defining identities

A note on normal division algebras of prime degree

a \cdot b = b \cdot a,\,\left[ {\left( {a \cdot a} \right) \cdot } \right] \cdot a = \left( {a \cdot a} \right) \cdot \left( {b \cdot a} \right)

On partially stable algebras

(1) The algebra \( {\mathfrak{A}^{\left( + \right)}} \) and its subalgebras are called special Jordan algebras.