Larders from Von Neumann Regular Rings

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.