Recent Progress on Going-Down I

Dobbs, David E.

doi:10.1007/978-1-4757-3180-4_7

Non-Noetherian Commutative Ring Theory 2000

DOI: 10.1007/978-1-4757-3180-4_7

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Recent Progress on Going-Down I

David E. Dobbs

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2021

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Dp-Minimal integral domains

d’Elbée

Halevi

2021

Isr. J. Math.

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We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from the following construction: there is a dp-minimal valuation overring O ⊇ R, a proper ideal I in O, and a finite subring R 0 ⊆ O/I such that R is the preimage of R 0 in O. Preliminaries NotationWe employ fairly standard model theoretic notation and conventions, see [TZ12,Sim15].We briefly review the definition of dp-rank and dp-minimality. If Σ(x) is a partial type and κ is a cardinal, an ict-pattern of depth κ in Σ(x) is an array of formulas (ϕ α (x, b α,i ) : α < κ, i < ω) over an elementary extension, such that for every function η : κ → ω, the type Σ(x) ∪ {ϕ α (x, b α,i ) : i = η(α)} ∪ {¬ϕ α (x, b α,i ) : i = η(α)}

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Dp-Minimal integral domains

d’Elbée

Halevi

2021

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

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Recent Progress on Going-Down I

Cited by 1 publication

References 56 publications

Dp-Minimal integral domains

Dp-Minimal integral domains

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