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Lattice-ordered Rings

In: Lattice-ordered Rings and Modules

Author

Listed:
  • Stuart A. Steinberg

    (University of Toledo, Department of Mathematics)

Abstract

Lattice-ordered rings occur as polynomial rings, power series rings, and semigroup rings, as do the perhaps more familiar totally ordered rings, but they also occur as matrix rings and endomorphism rings. We are concerned with the theory and structure of lattice-ordered rings and lattice-ordered modules and consequently a diverse number of topics appears. After initially supplying examples and identifying interesting classes of these objects we present the theory of radicals in the variety of lattice-ordered rings. Additional examples are provided when generalized power series rings are studied. As in ring theory radicals are connected with the structure of a lattice-ordered ring by means of the intent to factor out a bad radical and end up with a good quotient. The most useful radical comes from the class of nilpotent lattice-ordered rings. For this radical the good quotient has no nilpotent kernels and in the right situation it even lacks positive nilpotent elements. By concentrating on the variety of f -rings, which is the variety generated by the class of totally ordered rings, we are able to obtain more fruitful structural results. For example, in this variety the building blocks for the l-ring analogue of the Jacobson semisimple rings are the l-simple totally ordered domains with an identity element. This is also true for some varieties larger than the f-ring variety, but no such definitive identification is likely in general.

Suggested Citation

  • Stuart A. Steinberg, 2010. "Lattice-ordered Rings," Springer Books, in: Lattice-ordered Rings and Modules, edition 0, chapter 0, pages 125-279, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-1721-8_3
    DOI: 10.1007/978-1-4419-1721-8_3
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