Idealtheorie in Ringbereichen¶
Noether, E. (1921). Idealtheorie in Ringbereichen. Mathematische Annalen, 24-66.
Cited by¶
2 citations across 2 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Associativity
- and extended by Noether (1921) in commutative ring theory where the associative axiom is foundational
This sourceGives the modern axiomatic definition of a commutative ring (associative axiom foundational) and proves the primary-decomposition theorem; widely regarded as the founding paper of commutative ring theory.
- and extended by Noether (1921) in commutative ring theory where the associative axiom is foundational
- Commutativity
- The signature is minimal — just the operation and the axiom — but its consequences are wide: (a) expressions are invariant under input swapping, so a single operation result applies regardless of input order; (b) commutative binary operations extend naturally to \(n\)-ary operations on any finite multiset (the result is independent of enumeration order, so \(\sum_{i \in I} a_i\) and \(\prod_{i \in I} a_i\) are well-defined for finite index sets \(I\) without choosing an ordering); © the structure \((S, \circ)\) admits a simpler theory than non-commutative analogues (abelian group theory, commutative ring theory, commutative algebra), as Noether (1921) established in foundational work on ideal theory in commutative rings
This sourceFounding paper of commutative ring theory (ascending chain condition, primary decomposition; modern definition of a commutative ring) — supports the claim that Noether established the foundational theory of commutative rings.
- The signature is minimal — just the operation and the axiom — but its consequences are wide: (a) expressions are invariant under input swapping, so a single operation result applies regardless of input order; (b) commutative binary operations extend naturally to \(n\)-ary operations on any finite multiset (the result is independent of enumeration order, so \(\sum_{i \in I} a_i\) and \(\prod_{i \in I} a_i\) are well-defined for finite index sets \(I\) without choosing an ordering); © the structure \((S, \circ)\) admits a simpler theory than non-commutative analogues (abelian group theory, commutative ring theory, commutative algebra), as Noether (1921) established in foundational work on ideal theory in commutative rings
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