The minimal number of generators of an invertible ideal

Minimal Prüfer-Dress rings and products of idempotent matrices

“…However, the property of being Noetherian is specific of the Dress ring of R(X). In fact, by Schülting's result in [15], generalized by Olberding and Roitman in [12], if A contains more than one indeterminate, then D K contains finitely generated ideals that are not 2-generated. Therefore D K cannot be Noetherian, otherwise, being a Prüfer domain, it should be Dedekind, hence all its ideals should be generated by two elements.…”

“…. , X n ) of R K cannot be generated by less then n + 1 elements (see [15] and [12]). This result is deep and difficult to prove: techniques of algebraic geometry are required.…”

“…For K = R(A), A a set of indeterminates, we also examine the relations between D K and the Prüfer domains investigated by Schülting [15] (1979). Recall that Schülting was the first to prove the existence of Prüfer domains R containing finitely generated ideals that are neither principal nor two-generated (see also [12] for a neat discussion on this subject). As a by-product of the results in [15], we get that also some minimal Dress rings admit n-generated ideals that are not 2-generated.…”

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“…The fact that such rings are Prüfer has a number of interesting consequences for real algebraic geometry and sums of powers of elements of F; see, for example, [Becker 1982;Schülting 1982]. These rings are also the only known source of Prüfer domains having finitely generated ideals that cannot be generated by two elements, as was shown by Schülting [1979] and Swan [1984]; the related literature on this aspect of holomorphy rings is discussed in [Olberding and Roitman 2006]. The notion of existential closure leads to more general results on Prüfer holomorphy rings in function fields.…”

“…(4) In [Olberding and Roitman 2006] it is shown that if the holomorphy ring A of Z contains a field of cardinality greater than that of Z , then A is a Bézout domain.…”