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 a fortiori 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.
Read more about this topic: Noetherian Module
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