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... Unfortunately, they do not usually present a coherent class of these domains, or some sort of characterization of rings which are non-Euclidean PIDs in some distinguished class of domains. Nice attemps in this direction can be found in [1] and [6]. ...
Using a nonstandard model of Peano arithmetic, we show that there are quasi-Euclidean subrings of Q[x] which are not k-stage Euclidean for any norm and positive integer k. These subrings can be either PID or non-UFD, depending on the choice of parameters in our construction. In both cases, there are such domains up to ring isomorphism.
... Let (A, M ) be a two-dimensional regular local ring, and let 0 = f ∈ M 2 be a principal prime (e.g., A = K[[X, Y ]], K a field, and f = X 2 + Y 3 ). Then A f is a non-Euclidean PID [4]. Hence there is a proper ideal I of ...
Let R be a commutative ring with identity. For a, b ∈ R define a and b to be associates, denoted a ∼ b, if a\b and b\a, to be strong associates, denoted a ≈ b, if a = ub for some unit u of R, and to be very strong associates, denoted by a ≅ b, if a ∼ b and further when a ≠ 0, a = rb implies that r is a unit. Certainly a ≅ b ⇒ a ≈ b ⇒ a ∼ b. In this paper we study commutative rings R, called strongly associate rings, with the property that for a, b ∈ R, a ∼ b implies a ≈ b and commutative rings R, called présimplifiable rings, with the property that for a, b ∈ R, a ∼ b (or a ≈ b) implies that a ≅ b.
A classical problem, originated by Cohn's 1966 paper [1], is to characterize the integral domains R satisfying the property: (GEn) “every invertible n×n matrix with entries in R is a product of elementary matrices”. Cohn called these rings generalized Euclidean, since the classical Euclidean rings do satisfy (GEn) for every n>0. Important results on algebraic number fields motivated a natural conjecture: a non-Euclidean principal ideal domain R does not satisfy (GEn) for some n>0. We verify this conjecture for two important classes of non-Euclidean principal ideal domains: (1) the coordinate rings of special algebraic curves, among them the elliptic curves having only one rational point; (2) the non-Euclidean PID's constructed by a fixed procedure, described in Anderson's 1988 paper [2].