Logic Seminar
Thomas Kucera
University of Manitoba
Elementary socles and radicals
Abstract: 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 definable subgroups, the
elementary socle is non trivial and is a definably closed submodule. Furthermore, the definition of
elementary socle naturally extends 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 elementary 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 formulas in the languages of left
modules and right modules which Herzog extended to a duality of categories. This duality makes
modules of the kind described above correspond; and their internal structure is shown to be similar by
means of this duality.
This is a preliminary report on work in progress.
Monday April 3, 2006 at 4:00 PM in SEO 427