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Let R be a valuation domain. We investigate the notions of E(R)- algebra and generalized E(R)-algebra and show that for wide classes of maximal valuation domains R, all generalized E(R)-algebras have rank one. As a by-product we prove if R is a maximal valuation domain of finite Krull dimension, then the two notions coincide. We give some examples of E(R)-algebras of finite rank that are decomposable, but show that over Nagata domains of small degree, the E(R)-algebras are, with one exception, the indecomposable finite rank algebras.
Goldsmith, B. & Zanardo, P. (2006). Generalised E-Algebras over valuation domains. Forum Mathematicum, vol. 18, pg. 1027-40. doi:10.1515/FORUM.2006.052