Let D be a commutative domain with identity, and let $${\mathcal {L}}(D)$$ L ( D ) be the lattice of nonzero ideals of D . Say that D is ideal upper finite provided $${\mathcal {L}}(D)$$ L ( D ) is upper finite, that is, every nonzero ideal of D is contained in but finitely many ideals of D . Now let $$\kappa >2^{\aleph _0}$$ κ > 2 ℵ 0 be a cardinal. We show that a domain D of cardinality $$\kappa $$ κ is ideal upper finite if and only if D is a Dedekind domain. We also show (in ZFC) that this result is sharp in the sense that if $$\kappa $$ κ is a cardinal such that $$\aleph _0\le \kappa \le 2^{\aleph _0}$$ ℵ 0 ≤ κ ≤ 2 ℵ 0 , then there is an ideal upper finite domain of cardinality $$\kappa $$ κ which is not Dedekind.

中文翻译：

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大型 Dedekind 域的表征

令 D 为具有恒等式的可交换域，令 $${\mathcal {L}}(D)$$ L ( D ) 为 D 的非零理想格。假设 D 是理想的上有限条件，只要 $${\mathcal {L}}(D)$$ L ( D ) 是上有限的，也就是说， D 的每个非零理想都包含在 D 的有限多个理想中。现在让 $$\kappa >2^{\aleph _0}$$ κ > 2 ℵ 0 成为基数。我们证明了基数 $$\kappa $$ κ 的域 D 是理想的上有限当且仅当 D 是戴德金域。我们还表明（在 ZFC 中）这个结果是尖锐的，因为如果 $$\kappa $$ κ 是一个基数，使得 $$\aleph _0\le \kappa \le 2^{\aleph _0}$$ ℵ 0 ≤ κ ≤ 2 ℵ 0 ，则存在一个理想的基数上有限域 $$\kappa $$ κ 不是戴德金。