Department of Mathematics
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This is an introductory lecture.

I will give a brief review of the first-order theory of modules over a fixed ring R. There is elimination of quantifiers down to positive primitive formulas ("pp-elimination of quantifiers"). Positive primitive formulas define subgroups (but not necesarily submodules); positive primitive formulas with parameters, if consistent, define cosets of definable subgroups.

The set of all definable subgroups of a module is a sublattice of the lattice of subgroups; hence is a modular lattice with 0 and 1.

Definable chain conditions are expressed in terms of chain conditions on the lattice of pp-definable subgroups. Two stages in the "stability" classification of theories of modules are captured by this sort of chain condition.

Subsequent lectures will be devoted to an overview of recent work by Philipp Rothmaler of CUNY and myself on definable chain conditions.