Noetherian rings from a non-Noetherian perspective

Bruce Olberding
New Mexico State University



After Emmy Noether introduced the axiom of the ascending chain condition in 1921, commutative algebra developed very swiftly. From 1921 to the present, much of the impetus for this rapid growth has been to provide algebraic explanations and proofs of geometric facts; in turn, through the work of Grothendieck and many others, geometry proved useful in understanding algebraic ideas.  So successful and powerful were all these efforts that today commutative algebra is sometimes considered a chapter in algebraic geometry.  However, the ascending chain condition proves to be more robust than is indicated by this version of the story, and there persist classes of Noetherian rings resistant (so far) to the geometrical point of view.  We discuss some of these classes, with emphasis on analytically ramified local Noetherian rings and how techniques from non-Noetherian commutative ring theory are useful in understanding such rings.