Generalized $E$-Algebras via $\lambda$-Calculus I

by Goebel and Shelah. [GbSh:867]
Fundamenta Math, 2006
An R-algebra A is called E(R) --algebra if the canonical homomorphism from A to the endomorphism algebra End_R A of the R-module {}_R A, taking any a in A to the right multiplication a_r in End_R A by a is an isomorphism of algebras. In this case {}_R A is called an E(R) --module. E(R)-algebras come up naturally in various topics of algebra, so it's not surprising that they were investigated thoroughly in the last decade. Despite some efforts it remained an open question whether proper generalized E(R)-algebras exist. These are R --algebras A isomorphic to End_R A but not under the above canonical isomorphism, so not E(R) --algebras. This question was raised about 30 years ago (for R= Z) by Phil Schultz and we will answer it. For PIDs R of characteristic 0 that are neither quotient fields nor complete discrete valuation rings - we will establish the existence of generalized E(R)-algebras. It can be shown that E(R)-algebras over rings R that are complete discrete valuation rings or fields must trivial (copies of R). The main tool is an interesting connection between lambda-calculus (used in theoretical computer sciences) and algebra. It seems reasonable to divide the work into two parts, in this paper we will work in V=L (Godels universe) hence stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper.


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