On representability of *-regular rings and modular ortholattices
Florence Micol · Technischen Universität Darmstadt · 2008
In this thesis a proof is given that simple modular ortholattices possessing a chain with at least five elements (or four if they are arguesian) are coordinatizable by a *-regular ring also with respect to the orthocomplementation. This is based on the fact that their lattice reduct possesses a large partial three-frame and hence satisfies a stricter condition of coordinatization yielding the involution on the coordinatizing ring. Simple modular ortholattices play an important role in the equational theory of modular ortholattices, since any variety of modular ortholattices is generated by its simple members, as shown by Herrmann and Roddy. As a second main result, a characterization of the smallest class V of *-regular rings containing the class A of artinian *-regular rings and closed under homomorphic images (H), products (P) and regular substructures (S_r) is set up. In fact, the elements of V are exactly the *-regular rings that can be embedded into an atomic *-regular ring, resp. the *-regular rings that can be embedded into a product of rings of endomorphisms of some vector spaces with scalar product such that the involution in the regular ring corresponds to the adjunction of endomorphisms. Finally, V is obtained as S_r H S_r P A.