The best answer I've ever been able to come up with is that the class of noetherian rings contains the classical number rings $\mathbf{Z}$ and $\mathbf{R}$ and is closed under the formation of polynomial rings, localization, completion, and quotients. So it contains many of the rings you will come across in ordinary situations (whatever that means). It also has the advantage that the definition is tractable enough that if someone hands you an explicit ring, it's not out of the question to try to work out from scratch whether it's noetherian. If you're the kind of person who likes abstract fields, then they're also included.

On the other hand, I don't think of it as a truly fundamental concept, like say finite presentation. But there is no denying it's convenience. If you need to avoid some infinitary phenomena but you still want a broad class of rings, it's often hard to beat noetherianness. It's also quite good in situations where you're too lazy to work out exactly what finiteness conditions you care about.

The best answer I've ever been able to come up with is that the class of noetherian rings contains the classical number rings $\mathbf{Z}$ and $\mathbf{R}$ and is closed under the formation of polynomial rings, localization, completion, and quotients. So it contains many of the rings you will come across in ordinary situations (whatever that means). It also has the advantage that the definition is tractable enough that if someone hands you an explicit ring, it's not out of the question to try to work out from scratch whether it's noetherian. If you're the kind of person who likes abstract fields, then they're also included.

On the other hand, I don't think of it as a truly fundamental concept, like say finite presentation. But there is no denying its convenience. If you need to avoid some infinitary phenomena but you still want a broad class of rings, it's often hard to beat noetherianness. It's also quite good in situations where you're too lazy to work out exactly what finiteness conditions you care about.

The best answer I've ever been able to come up is that the class of noetherian rings contains the classical number rings $\mathbf{Z}$ and $\mathbf{R}$ and is closed under the formation of polynomial rings, localization, completion, and quotients. So it contains many of the rings you will come across in ordinary situations (whatever that means). It also has the advantage that the definition is tractable enough that if someone hands you an explicit ring, it's not out of the question to try to work out from scratch whether it's noetherian. If you're the kind of person who likes abstract fields, then they're also included.

On the other hand, I don't think of it as a truly fundamental concept, like say finite presentation. But there is no denying it's convenience. If you need to avoid some infinitary phenomena but you still want a broad class of rings, it's often hard to beat noetherianness. It's also quite good in situations where you're too lazy to work out exactly what finiteness conditions you care about.

The best answer I've ever been able to come up with is that the class of noetherian rings contains the classical number rings $\mathbf{Z}$ and $\mathbf{R}$ and is closed under the formation of polynomial rings, localization, completion, and quotients. So it contains many of the rings you will come across in ordinary situations (whatever that means). It also has the advantage that the definition is tractable enough that if someone hands you an explicit ring, it's not out of the question to try to work out from scratch whether it's noetherian. If you're the kind of person who likes abstract fields, then they're also included.

On the other hand, I don't think of it as a truly fundamental concept, like say finite presentation. But there is no denying it's convenience. If you need to avoid some infinitary phenomena but you still want a broad class of rings, it's often hard to beat noetherianness. It's also quite good in situations where you're too lazy to work out exactly what finiteness conditions you care about.