  An R-algebra A is called E(R)-algebra if the canonical homomorphism from A to
the endomorphism algebra End_RA 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
(Godel 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.
