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Larders from Von Neumann Regular Rings

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2029)

Abstract

The assignment that sends a regular ring R to its lattice of all principal right ideals can be naturally extended to a functor, denoted by L (cf. Sect. 1.1.2). An earlier occurrence of a condensate-like construction is provided by the proof in Wehrung (J. Math. Log. 6(1):1–24, 2006, Theo- rem 9.3). This construction turns the non-liftability of a certain 1-lattice endomorphism from M ω (cf. Example 1.1.9) to a non-coordinatizable, 2-distributive complemented modular lattice, of cardinality 1, with a spanning M ω. Thus the idea to adapt the functor L to our larder context is natural. The present chapter is designed for this goal. In addition, it will pave the categorical way for solving, in the second author’s paper (Wehrung, A non-coordinatizable sectionally complemented modular lattice with a large J’onsson four-frame, Adv. in Appl. Math., to appear. Available online at http://hal.archives-ouvertes.fr/hal-00462951), a 1962 problem by J’onsson.

Keywords

  • Regular Ring
  • Ring Homomorphism
  • Modular Lattice
  • Lattice Isomorphism
  • Idempotent Element

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Pierre Gillibert .

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© 2011 Springer-Verlag Berlin Heidelberg

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Gillibert, P., Wehrung, F. (2011). Larders from Von Neumann Regular Rings. In: From Objects to Diagrams for Ranges of Functors. Lecture Notes in Mathematics(), vol 2029. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21774-6_6

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