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Noether connected finite ideal generation with the ascending-chain condition and obtained finite primary decomposition for ideals in commutative rings satisfying this condition. Finite irreducible decompositions provide the bridge: under the finiteness hypothesis irreducible ideals are primary. This lets algebraic objects be analyzed through a finite collection of components. Noether explicitly builds on earlier work of Lasker and Dedekind.
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Scientific scope
The finiteness condition is essential. Primary components need not be unique, and a primary ideal need not be a power of a prime. This does not assert decomposition in arbitrary rings.
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People and systems
Emmy Noether
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English catalogue narrative
Noether: finite primary decomposition under an ideal finiteness condition
Noether connected finite ideal generation with the ascending-chain condition and obtained finite primary decomposition for ideals in commutative rings satisfying this condition.
Finite irreducible decompositions provide the bridge: under the finiteness hypothesis irreducible ideals are primary. This lets algebraic objects be analyzed through a finite collection of components. Noether explicitly builds on earlier work of Lasker and Dedekind.
Scientific scope
The finiteness condition is essential. Primary components need not be unique, and a primary ideal need not be a power of a prime. This does not assert decomposition in arbitrary rings.
People and systems
Emmy Noether