Kevin McCrimmon

On Herstein's theorems relating Jordan and associative algebras

In ([3]; also [5]) I. K. Herstein gave a construction associating with any Jordan ideal J in a Jordan algebra of the form A+ an associative ideal B in the associative algebra A. He used this to show that if A was simple, so was A-:-. In ([4]; also [5]) he gave an analogous construction associating with any Jordan idcal J in the Jordan algebra 11(/f, *) of *-symmetric elements an associative *-ideal B in the associative algebra A with involution *. This was used to show (in characteristic f 2) that if A was *-simple then H(A, *) was simple. The purpose of this paper is to show that his constructions can also be used to relate the radicals of A!or H(A, *) with that of A. Specifically me will prove

Seminormality and root closure in polynomial rings and algebraic curves

Let D be an integral domain with identity and let K be the quotient field of D. Then D is said to be root closed if whenever 01 E K with (Y~ E D for some positive integer n, then 01 E D. The domain D is called (2, 3)-closed if whenever 01 E K with a2, a3 E D, then OL E D and D is called F-closed if whenever OL E K with nar E D for some positive integer n and OLD, 01~ E D, then ol E D. Clearly, if D is root closed, then D is (2,3)-closed and if D is (2,3)-closed, then D is F-closed. The property of being root closed arose in Sheldon’s work [7] on how changing D changes the quotient field of D[[Xj]. As for (2, 3)-closure, its significance is due to the fact that an integral domain D is (2, 3)-closed if and only if D is seminormal if and only if Pit(D) = Pic(D[X, ,..., X,]), where Pit denotes the Picard group [4, Theorem 11. Concerning F-closure, it is shown in [l] that the domain D is F-closed if and only if D[X’j is D-invariant-i.e., if and only if whenever (D[Xj) [XI ,..., X,l ho 5’[Y, ,..., Yn], then S is D-isomorphic to D[x]. As noted in the preceding paragraph, these properties are related, although in their original non-arithmetic forms, they did not appear to be. It is a consequence, albeit a deeply hidden one, of the proof of Theorem 1 of [4] that if D is (2,3)-closed, so is D[Xj. In this paper, we give arithmetic proofs that each of these three properties respects polynomial extension. In fact, we do this in greater generality and our approach is a unifying one in that we show that if an integral domain D is “n-root closed”, then so is D[Xj. The notion of ‘%-root closed” is then utilized to delineate the arithmetic distinction between normality and seminormality for algebraic curves. Recall that Bombieri [2] has shown that the geometric distinction between a normal curve and a seminormal curve is that the seminormal curve may have “ordinary” singular points. We prove that the coordinate ring of an irreducible algebraic curve over an algebraically closed field K is integrally closed if and only if it is (2, 3)-closed plus n-root closed for some n prime to the characteristic of K. In the process of showing that our results cannot be extended to arbitrary reduced rings, we give a negative answer to a question implicit in [5, Ex.11,

A characterization of the radical of a Jordan algebra

The Jacobson radical of a Jordan algebra has been defined [5] as the maximal ideal consisting entirely of quasi-invertible elements. In this paper we shall obtain a characterization of the radical as the set of properly quasiinvertible elements, in analogy with the case of associative algebras. An element is properly quasi-invertible if it is quasi-invertible in all homotopes. (We show this characterization also works in the associative case). We apply our characterization of the radical of J to describe the radical of U,J, e an idempotent in J, and the radical of an ideal R C J. Throughout we use the notations and terminology of [4] for quadratic Jordan algebras over an arbitrary ring of scalars @. We recall the basic axioms for the composition U,y in the case of unital algebras:

An elemental characterization of strong primeness in Jordan systems

Abstract We give elemental characterizations of strong primeness for Jordan algebras, pairs and triple systems. We use our characterization to study the transfer of strong primeness between a Jordan system and its local algebras and subquotients.

The Zelmanov approach to Jordan homomorphisms of associative algebras

Abstract Zelmanovs work on prime Jordan algebras leads to an idempotent-free version of Martindales Theorem on the extension of Jordan homomorphisms and derivations from the hermitian elements H(R, *) of an associative algebra of degree >2 with involution to associative homomorphisms and derivations on R. The condition that J = H(R, *) be of degree >2 is replaced by the intrinsic condition that J = Z(J) ⊂ H(A,*) consist entirely of values of Zelmanov polynomials.

Zelmanov's prime theorem for quadratic Jordan algebras

E. I. Zelmanov has recently shown that any prime linear Jordan algebra without nil ideals (but no finiteness conditions imposed) is either a homomorphic image of a special algebra or is a form of the 27-dimensional exceptional algebra. In particular, all division algebras and all simple unital algebras are either derived from associative algebras or are 27.dimensional over their centers. This effectively ends the search for infinite-dimensional exceptional algebras. In the present paper, Zelmanov’s result is extended to quadratic Jordan algebras. Jordan algebras were invented in the 1930s in the search for an exceptional algebraic setting for quantum mechnics: an algebraic system which behaved like the usual algebra of quantum mechanical observables (hermitian operators on Hilbert space), but was exceptional in the sense that its structure was not determined behind the scenes by some unobservable associative algebra. In their pioneering paper of 1933, [ 91 Jordan, von Neumann, and Wigner classified the finite-dimensional formally real linear Jordan algebras, and A. A. Albert showed [ 11 that the only simple algebra in the list which was exceptional was a certain 27.dimensional algebra of 3 X 3 hermitian matrices with entries from an gdimensional Cayley algebra. All subsequent investigations of exceptional Jordan algebras led back to this same Albert algebra. The next breakthrough in the structure of Jordan algebras was N. Jacobson’s 1965 introduction [6 J of inner ideals and the resultant quadratilication of the theory (in terms of the quadratic product xyx instead of the linear product xy + .vx). The final breakthrough was the astounding 1979 result of a young Russian mathematician, E. I. Zelmanov, that the only simple exceptional linear Jordan are Albert algebras. Thus the search for an infinite-dimensional setting for exceptional quantum mechanics is doomed to failure: the only simple exceptional structure allotted to mortals is the 27-dimensional Albert algebra. This came as a complete surprise to Western researchers (who hoped the free Jordan algebra might lead to an 291 002 I-8693/82/060297-30

The kantor doubling process revisited

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A characterization of the Jacobson-Smiley radical

A follow-up on the authors′ earlier discussion on the Kantor construction of Jordan superalgebras is presented. A short axiom, required in characteristic 3 but omitted in the original treatise, is added to those of a Jordan superbracket. A particular case of the Kantor construc¬tion is the Jordan superalgebra of Poisson Brackets. Jordan folklore describes this superalgebra in Clifford terms, but we show this description does not even yield a Jordan superalgebra. An amended Clifford description of the Poisson superalgebra is given which does represent the superalgebra of Poisson brackets.

Coordinatization of triangulated Jordan systems

The Jacobson-Smiley radical rad ‘2l of an alternative algebra ?l is the maximal ideal consisting entirely of elements z which are quasi-invertible in the sense that 1 x is invertible. It has long been conjectured that, as in the associative case, the radical consists precisely of the elements z which are properly quasi-invertibIe in the sense that all az and %a (a E

Generic Jordan Polynomials

3) are quasiinvertible. In this note we will establish this conjecture, We also show that proper quasi-invertibility of z is equivalent to the condition that s be quasi-invertible in all homotopes ‘