### 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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