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This question already has an answer here:

While in undergraduate years, abstract stuctures are very confusing since they arise without many motivations most of the times, and find their living ground later on. I would like to understand why noetherian rings are interesting.

Principal and euclidean rings are definitely useful to have arithmetic tools and notion generalizing those of the arithmetics of integers. What about noetherian rings? It often appears in arithmetic courses, however what allows them to do that other rings do not? What are the lost properties compared to a principal ring?

In one sentence: how to grasp the meaning and interest of noetherian rings (or other structures)?

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marked as duplicate by rschwieb abstract-algebra Oct 5 '18 at 13:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Hello: please type your question into the search box before posting. Quite often you will find existing versions of the questions with good answers. If you discover you have a followup after reading such a question, you could go about searching/asking for that one. $\endgroup$ – rschwieb Oct 5 '18 at 13:17
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Noetherian conditions bring finiteness to bear, which helps a lot.

Consider the rich theory of finite dimensional vector spaces versus the not-so-rich theory of general vector spaces, for instance.

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Noetherian rings are important because essentially every example motivating rings, namely the rings naturally considered by Number Theory and Algebraic Geometry is noetherian.

The noetherianity condition, which is usually given as the stationarity of ascending chain of ideals but in fact equivalent to the finite generation of every ideal, is a natural one encompassing all said examples and has some very important implications (to name just one, the existence of maximal ideals and thus of quotient fields).

All of this is enough motivation to make the class of noetherian rings a very important one.

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Noetherian rings have a stopping condition in the form that every ascending chain of ideals becomes stationary. This is particularly interesting in computations. For instance, the polynomial ring ${\Bbb K}[x_1,\ldots,x_n]$ is noetherian which follows from the Hilbert basis theorem. This can be used to show that the Buchberger algorithm for the construction of a Gröbner basis of an ideal terminates.

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