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      "statement": "# Noether: finite primary decomposition under an ideal finiteness condition\n\nNoether connected finite ideal generation with the ascending-chain condition and obtained finite primary decomposition for ideals in commutative rings satisfying this condition.\n\nFinite 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.\n\n## Scientific scope\n\nThe 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.\n\n## People and systems\n\nEmmy Noether\n\n\n\n## Original sources\n\n- [Idealtheorie in Ringbereichen (1921); English translation by Daniel Berlyne (2014)](https://arxiv.org/abs/1401.2577)\n",
      "summary": "Noether connected finite ideal generation with the ascending-chain condition and obtained finite primary decomposition for ideals in commutative rings satisfying this condition.\n\nFinite 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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