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norajill
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Is Z a noetherian Z-module
Yes, Z is a noetherian Z-module. This means that every submodule of Z is finitely generated. In other words, any subset of Z can be written as a finite linear combination of elements in Z.
A noetherian Z-module is a module over the ring of integers, Z, that satisfies the ascending chain condition. This means that any increasing chain of submodules will eventually stabilize and become constant. In simpler terms, every submodule can be generated by a finite number of elements.
The noetherian property is useful in Z-modules because it allows for the study of finitely generated modules, which are easier to understand and work with. It also allows for the use of important theorems and techniques, such as the Hilbert basis theorem, to prove results about these modules.
Yes, a Z-module can be noetherian over a different ring. The noetherian property is not specific to the ring of integers and can be applied to modules over any ring. However, the definition and properties may vary depending on the ring.
Some examples of noetherian Z-modules include the ring of integers itself, polynomial rings over Z, and finite abelian groups. Any finitely generated Z-module will also be noetherian.