Chapter

Algebraic Geometry and Commutative Algebra

Part of the series Universitext pp 55-82

The Theory of Noetherian Rings

  • Siegfried BoschAffiliated withMathematisches Institut, Westfälische Wilhelms-Universität

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Abstract

A ring is called Noetherian if all its ideals are finitely generated or, equivalently, if its ideals satisfy the ascending chain condition. The aim of the chapter is to show that the Noetherian hypothesis, as simple as it might look, nevertheless has deep impacts on the structure of ideals and their inclusions, such as the existence of primary decompositions and, as a culminating point, the theory of Krull dimension.