Editing Noetherian ring (section)

=== Noetherian group rings ===
Consider the [[group ring]] <math>R[G]</math> of a [[group (mathematics)|group]] <math>G</math> over a [[ring (mathematics)|ring]] <math>R</math>. It is a [[ring (mathematics)|ring]], and an [[associative algebra]] over <math>R</math> if <math>R</math> is [[commutative ring|commutative]]. For a group <math>G</math> and a commutative ring <math>R</math>, the following two conditions are equivalent.
* The ring <math>R[G]</math> is left-Noetherian.
* The ring <math>R[G]</math> is right-Noetherian.
This is because there is a [[bijection]] between the left and right ideals of the group ring in this case, via the <math>R</math>-[[associative algebra]] [[homomorphism]]
:<math>R[G]\to R[G]^{\operatorname{op}},</math>
:<math>g\mapsto g^{-1}\qquad(\forall g\in G).</math>
Let <math>G</math> be a group and <math>R</math> a ring. If <math>R[G]</math> is left/right/two-sided Noetherian, then <math>R</math> is left/right/two-sided Noetherian and <math>G</math> is a [[Noetherian group]]. Conversely, if <math>R</math> is a Noetherian commutative ring and <math>G</math> is an [[group extension|extension]] of a [[Noetherian group|Noetherian]] [[solvable group]] (i.e. a [[polycyclic group]]) by a [[finite group]], then <math>R[G]</math> is two-sided Noetherian. On the other hand, however, there is a [[Noetherian group]] <math>G</math> whose group ring over any Noetherian commutative ring is not two-sided Noetherian.<ref name="Ol’shanskiĭ">{{cite book
|last1=Ol’shanskiĭ
|first1=Aleksandr Yur’evich
|title=Geometry of defining relations in groups
|translator-last=Bakhturin
|translator-first=Yu. A.
|language=en
|series=Mathematics and Its Applications. Soviet Series
|volume=70
|publisher=Kluwer Academic Publishers
|location=Dordrecht
|date=1991
|isbn=978-0-7923-1394-6
|issn=0169-6378
|doi=10.1007/978-94-011-3618-1
|mr=1191619
|zbl=0732.20019
}}</ref>{{rp|423, Theorem 38.1}}