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. Verified; DOI resolves to the correct article.
Supported in partVerified against the publisher's abstract
Noether 1921 gives an axiomatic ring definition using the associative law, but this abstract does not state that the associative axiom is foundational in commutative ring theory.
“Die Definition ist der Fraenkelschen Habilitationsschrift entnommen, unter Weglassung dessen einschränkender Bedingungen 6, I und II; dafür mußte das kommutative Gesetz der Addition mit aufgenommen werden.”
- 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.
Supported in partVerified against the publisher's abstract
The work is Noether 1921 on ideal theory in rings, but the abstract we hold states no comparative claim about commutative versus non-commutative theories.
“Die Existenz der Zerlegung läßt sich wieder in Analogie zu § 2 direkt beweisen; auch der Eindeutigkeitsbeweis läßt sich direkt führen (vgl. das in der Einleitung über Schmeidler und Noether-Schmeidler Gesagte).”
- 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
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 2 of 2 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.
See how references were verified.
Registry ID ref:ad1b23a4ed9f · see in the full table