In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma.

PropertyValue
dbpprop:abstract
  • In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory. Although the various prime ideal theorems may appear simple and intuitive, they cannot be derived in general from the axioms of Zermelo–Fraenkel set theory (ZF). Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others—the Boolean prime ideal theorem, for instance—represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF+AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.
  • Twierdzenie o ideale pierwszym – twierdzenie teorii krat rozdzielnych.
  • 在数学中，素理想定理保证在给定的抽象代数中特定类型的子集的存在性。常见的例子就是布尔素理想定理，它声称在布尔代数中的理想可以被扩展成素理想。这个陈述对于在集合上的滤子的变体叫做叫做超滤子引理。通过考虑不同的带有适当的理想概念的数学结构可获得其他定理，例如环和(环论的)素理想，和分配格和的极大理想。本文关注序理论的素理想定理。 尽管各种素理想定理可能看起来简单和直觉性的，它们一般不能从 Zermelo–Fraenkel 集合论(ZF)的公理推导出来。反而某些陈述等价于选择公理(AC)，而其他的如布尔素理想定理，体现了严格弱于 AC 的性质。由于这个在 ZF 和 ZF+AC (ZFC) 之间的中介状态，布尔素理想定理经常被接受为集合论的公理。经常用缩写 BPI(对布尔代数)或 PIT 提及这个额外公理。
dbpprop:hasPhotoCollection
rdfs:comment
  • In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma.
  • Twierdzenie o ideale pierwszym – twierdzenie teorii krat rozdzielnych.
rdfs:label
  • Boolean prime ideal theorem
  • Twierdzenie o ideale pierwszym
  • 布尔素理想定理
owl:sameAs
skos:subject
foaf:page
is dbpprop:redirect of