# Why noetherian rings [duplicate]

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)?

## marked as duplicate by rschwieb abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 5 '18 at 13:16

• 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. – rschwieb Oct 5 '18 at 13:17

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.