 
Summary: UIC Model Theory Seminar, April 3, 2006
Elementary socles and radicals
Thomas Kucera
University of Manitoba
The elementary socle of a module is the sum of all the minimal nonzero first
order definable subgroups of that module. Dually the elementary radical of
a module is the intersection of all the maximal proper firstorder definable
subgroups of that module. These concepts were first introduced by Ivo
Herzog in his thesis.
If an indecomposble module has the descending chain condition on defin
able subgroups, the elementary socle is non trivial and is a definably closed
submodule. Furthermore, the definition of elementary socle naturally ex
tends to an ascending series of definably closed submodules whose union is
the whole module. Dually, if an indecomposable module is pureinjective
and has the ascending chain condition on definable subgroups, the elemen
tary radical is a submodule, and the definition of the elementary radical may
be extended to a descending series of submodules whose intersection is 0.
The definitions and some of the properties generalize in natural ways to
arbitrary (indecomposable) pureinjective modules.
Mike Prest introduced a notion of duality between certain first order
