Editing Noetherian module (section)

==Use in other structures==
A right [[Noetherian ring]] ''R'' is, by definition, a Noetherian right ''R''-module over itself using multiplication on the right.  Likewise a ring is called left Noetherian ring when ''R'' is Noetherian considered as a left ''R''-module.  When ''R'' is a [[commutative ring]] the left-right adjectives may be dropped as they are unnecessary.  Also, if ''R'' is Noetherian on both sides, it is customary to call it Noetherian and not "left and right Noetherian".
 
The Noetherian condition can also be defined on [[bimodule]] structures as well: a Noetherian bimodule is a bimodule whose [[poset]] of sub-bimodules satisfies the ascending chain condition. Since a sub-bimodule of an ''R''-''S'' bimodule ''M'' is in particular a left ''R''-module, if ''M'' considered as a left ''R''-module were Noetherian, then ''M'' is automatically a Noetherian bimodule. It may happen, however, that a bimodule is Noetherian without its left or right structures being Noetherian.