SEMINAR: THE OSU-OU RING THEORY SEMINAR

TITLE:  Elementary socles and radicals

SPEAKER:  Thomas Kucera

                 University of Manitoba, Canada

DAY/DATE: FRIDAY, April 21, 2006

TIME: 4:45 P.M.

ROOM: MW 154

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 potentially most promising application is to the structure of

indecomposable injective (left) modules over (left) noetherian rings.

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.