Skeletons, bodies and generalized $E(R)$-algebras

Abstract

In this paper we want to solve a fifty year old problem on R-algebras over cotorsionfree commutative rings R with 1. For simplicity (but only for the abstract) we will assume that R is any countable principal ideal domain, but not a field. For example R can be the ring Z or the polynomial ring Q [x]. An R-algebra A is called a generalized E(R)-algebra if its algebra End R A of R-module endomorphisms of the underlying R-module R A is isomorphic to A (as an R-algebra). Properties, including the existence of such algebras are derived in various papers ([5, 6, 9, 10, 20, 22, 24, 25]). The study was stimulated by Fuchs [13], and specially by Schultz [26]. But due to [26] the investigation concentrated on ordinary commutative E(R)-algebras. A substantial part of problem 45 (p. 232) in the monograph [13] (repeated in later publications, e.g. [27]), which will be answered positively for all rings R above in this paper, remained open:Can we find non-commutative generalized E(R)-algebras?In Theorem 1.5 we will show that for all countable, principal ideal domains R which are not fields and for any infinite cardinal κ there is a non-commutative R-algebra A of cardinality |A| = κ ℵ 0 with End R A ∼ = A, so A is a non-commutative generalized E(R)-algebra, and-not too surprisingly-there is a proper class of examples.The new strategy should be interesting and useful for other problems as well: We will first translate the heart of the algebraic question on the existence of certain monoids via model theory into geometric structures leading to a special class of (decorated) trees and solve this problem introducing products of trees etc. This can be compared with the well-known, but different process of translating group problems to small cancelations in groups via the van Kampen lemma. By small cancelation of trees we are able to find a suitable monoid and thus a non-commutative algebra A with an important non-canonical embedding A → End R A, our * -scalar multiplication. In a second part of this paper we must enlarge A to get rid of all undesired endomorphisms. This can be done more easily (thus first) with the help of additional set theory (Jensen's diamond predictions), which will support the reader to understand more quickly the last steps of the proof. In a final chapter we will also give an argument (for removing unwanted endomorphisms) which is based on R. Göbel:Mathematics Subject Classification (2000): Primary 20K20, 20K30; Secondary 16S60, 16W20 GbHSh:943 in Shelah's list of publications 846 Rüdiger Göbel et al. ordinary set theory of ZFC only (using our favored Black Box predictions; see [20]). Thus we get the result as stated above. Furthermore, this last chapter includes a construction for rigid systems of non-commutative generalized E(R)-algebras. IntroductionAs indicated in the abstract we want to construct certain non-commutative R-algebras over commutative rings R with 1. Recall that an R-algebra A is a generalized E(R)algebra if End R A ∼ = A as R-algebras, where End R M denotes the algebra of R-module endomorphi...

Zeeman-Like Topologies in Special and General Theory of Relativity

“…In this section, we describe and discuss the work of Göbel [7,8], Lindstrom [11] and others on Zeeman-like topologies defined on a space-time of general relativity. In particular, Göbel [7] has proved the result that two space-times are homeomorphic with respect to its Zeeman topology if and only if they are isometric.…”

“…Modified results about Zeeman-and Zeeman-like topologies were published in the context of both special as well as general theory of relativity. Most remarkable are the results by S. Nanda [3,4,5], G. Williams [6], R. Göbel [7,8], Hawking-King-McCarty [9], Malament [10] and Lindstrom [11] proved in 1970's. S.G. Popvassilev [12] generalized some of these results to R n .…”

Basic Notions

“…The Lorentz metric, being non-positive definite, does not define any topology [3]: its role is actually to introduce causality. Many proposals have been made to fix that topology [4][5][6][7][8][9][10], but none has obtained general acceptance. In practice, physicists make implicitly a purely operational option: they use an underlying euclidean E 4 when eventually using global coordinates, or when talking about "continuous" fields.…”