Prime non-Lie modules for Mal'tsev superalgebras

A module V for a superalgebra A is called prime if any two of its nonzero submodules have a nonzero intersection, and no nonzero submodule is annihilated by a nonzero ideal of A. We prove that if V is a prime module for a Mal'tsev superalgebra M = M₀+M₁, one of the following cases is realized:

(1)	M0 = 0 and M1 consists of noneven mutually commuting injective endomorphisms of a -module V = V0 + V1, which is a prime module for an associative commutative Z2-graded algebra alg M1 EndV;
(2)	M1 = 0, the central closure Z–1 M of the algebra M = M0 either is a central, simple, seven-dimensional, non-Lie algebra or is a central, simple, three-dimensional, Lie algebra over a field Z–1 Z, and the central closure Z–1V of the module V = V0 is isomorphic to the (unique) non-Lie irreducible module over Z–1 M.

Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 448–465, July-August, 1994.