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Minimax-Moduln. (Minimax modules). (German) Zbl 0593.13012
The author studies modules M over a commutative Noetherian ring R with the property that M/N is Artinian for some Noetherian submodule N. By analogy with certain groups considered by R. Baer, he calls such objects minimax modules. - The author provides a number of alternative definitions. For example M is minimax if and only if in every ascending chain of submodules of M almost all the factors are Artinian. This follows from the more general result that a module M over R satisfies the maximal condition for submodules U for which the socle of M/U is trivial, if and only if M is Noetherian by semi-Artinian, where a semi-Artinian module is one that is generated by its Artinian submodules. Dually M is minimax if and only if in every descending chain of submodules of M almost all the factors are Noetherian. This, too, follows from a more general result. The author is led to consider modules M satisfying the minimal condition for submodules U for which the socle of M/U is trivial. He proves that M has this property if and only if M modulo the sum of its Artinian submodules has finite Goldie dimension and if dim(R/P)\(\leq 1\) for all \(P\in Ass(M)\). The paper also considers a number of other related concepts and analyses their relationships to each other and to the above.
Reviewer: B.A.F.Wehrfritz

13E99 Chain conditions, finiteness conditions in commutative ring theory
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13E05 Commutative Noetherian rings and modules
Full Text: DOI
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