Vol. 40, No. 3, 1972

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Finitely-valued f-modules

Stuart A. Steinberg

Vol. 40 (1972), No. 3, 723–737

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

Mathematical Subject Classification
Primary: 06A70
Received: 25 November 1970
Revised: 12 October 1971
Published: 1 March 1972
Stuart A. Steinberg