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Preprint Number 1820

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1820. Ivo Herzog and Sonia L'Innocente
The lattice coordinatized by a ring
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Submission date: 27 July 2020

Abstract:

A coordinatization functor L(-,1): Ring → Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functors are 1) that it extends the functor R → L(R) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective (-)op endofunctors on Ring and Latt; and 3) it admits localization at a left R-module. The complemented elements of L(R, 1) form a partially ordered set S(R) isomorphic to the space of direct summands of RR.

The right nonsingular rings for which the embedding of S(R) into the localization L(R, 1)Q at the right maximal ring of quotients QR is an isomorphism are characterized by a property that every finite matrix subgroup φl(RR) of the left R-module RR is essential in an element of S(R). In that case, the space S(R) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

Mathematics Subject Classification: 03C60, 06B15, 16S90.

Keywords and phrases: Modular lattice, coordinatization, positive primitive formula, the space of summands, (von Neumann) regular ring, maximal ring of quotients, ring of definable scalars, dominion.

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