I. Kaplansky discovered a simple 3-dimensional half-unital Jordan superalgebra kappa3(PHI) spanned over a field PHI by eo, xi1, eta1 with eo.xi1 = 1/2xi1, e0.eta1 = 1/2eta1, eta1.xi = e0. This Tiny Kaplansky superalgebra is special with specialization [GRAPHICS] in the 2x2 matrices M2(D) over the algebra D = PHI [L(t), d/dt] of differential operators on the polynomial algebra PHI[t]. In this paper we obtain a general class of half-unital Kaplansky superalgebras kappa(M) = kappa0(A) + kapp1(M) with kappa0(a).kappa0(b) = kappa0(ab), kappa0(a).kappa1(M) = 1/2kappa1(am), kappa1(m).kappa1(n) = kappa0(m x n) built from a bracket module M = (A, M, x) consisting of a scalar algebra A, a unital A-module M, and a skew product on M x M to A. These superalgebras are simple iff M is simple as bracket module iff A contains no proper M-invariant ideals (ideals invariant under all odd derivations D(m,n)(a) = (am) x n-a(m x n)) and M x M = A. kappa(M) is Jordan iff M satisfies a Jacobi identity SIGMA(cyclic) (m(i) X m(j))m(k) = 0 and a derivation identity 2a(m x n) = am x n + m x an. The two basic examples of bracket modules are nested in scalar algebras F: A = F0, M = F1 for F0 subset-of F1 subset-of F with F0 a unital subalgebra of F and F1 an F0-module with bracket product f x g = D(f ) g - fD(g) for D: F1 --> F0 a derivation of F0-modules. When D is a global derivation of F the Jordan superalgebra K(F0, F1, X D) is special with specialization [GRAPHICS] in the 2 x 2 matrices M2(D) over the algebra D = PHI < L(F1), D > on F. The first basic example is the Tiny Kaplansky module M3(PHI) (A = PHI1, M = PHI1 + PHIt, F = PHI[t], D = d/dt, kappa(M3(PHI)) = kappa3(PHI)), which is simple iff PHI is a field. The second basic example is the full derivation module M(F, D) (A = M = F, f x g = D(f ) g -fD(g) for a derivation D of F), which is simple iff F is D-simple, e.g., F the infinite-dimensional PHI[t] or PHI[[t]] for D=d/dt over a field of characteristic 0, or F the p-dimensional truncated polynomial algebra PHI[t(p)], D = d/dt(p) over a field of characteristic p. Our main result is that if M is simple of characteristic p > 0 (or is simple of characteristic 0 with A local, e.g., algebraic over a field), then M is either tiny or full; if M is simple of characteristic 0 then M imbeds in a full M(F, D) over a field F. In particular, all simple Kaplansky superalgebras kappa(M) are special. (C) 1994 Academic Press, Inc.
