Department of Mathematics

Rings and Modules Seminar


T. Kucera
Thomas.Kucera(at)Umanitoba(dot)CA
Department of Mathematics
University of Manitoba
Tuesday, October 22, 2013
Abstract:
This talk will be an introduction to and a very preliminary report on joint work being carried out with Philipp Rothmaler of the City University of New York. If \(I\) is a nonempty set an \(I\)property of modules is some operation \(P\) on the class of modules, which for each module \(M\) selects some subset \(P(M)\) of \(M^{I}\). Similarly, an \( (I) \)property \(Q\) selects some subset of \(M^{(I)}\). Imposing natural algebraic constraints on \(I\)properties force them to be representable by various kinds of infinitary pp formulas. As yet we have no natural semantics for \( (I) \)properties. Both kinds of properties naturally organize into lattices. For a fixed ring \(R\) there is a duality reflecting finitary first order properties (for finite \(I\)) between the lattice of properties of right \(R\)modules and the lattice of properties of left \(R\)modules. We attempt to generalize this "elementary duality" in various ways to infinitary properties of modules. These matters will be discussed in more detail in subsequent talks this fall and winter. 