Summary: UIC Model Theory Seminar, April 3, 2006
Elementary socles and radicals
University of Manitoba
The elementary socle of a module is the sum of all the minimal non-zero first-
order definable subgroups of that module. Dually the elementary radical of
a module is the intersection of all the maximal proper first-order 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 pure-injective
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) pure-injective modules.
Mike Prest introduced a notion of duality between certain first order