Finitely-valuedf-modules

Abstract

Let M be a right /-module over the directed po-ring R (i.e., M is a lattice-ordere d E-module that is a subdirect product of a family of totally ordered ϋJ-modules), and let g be a nonzero element of M. There is a natural one-to-one correspondence between the set of ϋί-values of g in M and the set of ^-values of g in M. This basic fact enables one to obtain all of the local structure theory for /-modules that Conrad [Czechoslovak Math. J. 15 (1965)] has obtained for /-groups. There is, in addition, the interaction between the two structures. For example, a special element g has the same value in Cn(g), the convex /-submodule generated by g, that it has in Cz(g). Using this structure theory and the fact that a special element is basic in a Johnson semisimple /-ring, it is shown that a finitely-valued Johnson semisimple /-ring is a direct sum of unital /-simple /rings.