12/08/14 07:17:20.69
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Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor.
By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite.
A Noetherian module is a module in which every strictly ascending chain of submodules breaks off after a finite number.
A Noetherian space is a topological space in which every strictly increasing chain of open subspaces breaks off after a finite number of terms; this definition is made so that the spectrum of a Noetherian ring is a Noetherian topological space.
The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space, are Noetherian themselves;
all subgroups and quotient groups of a Noetherian group are likewise, Noetherian; and, mutatis mutandis, the same holds for submodules and quotient modules of a Noetherian module.
All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings.
The chain condition also may be inherited by combinations or extensions of a Noetherian object.
For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
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