Artinian serial modules over commutative (or, left Noetherian) rings are at most one step away from being Noetherian. (English) Zbl 1375.16011
Summary: We introduce and study the concept of dual perfect dimension which is a Krull-like dimension extension of the concept of acc on finitely generated submodules. We observe some basic facts for modules with this dimension, which are similar to the basic properties of modules with Noetherian dimension. For Artinian serial modules, we show that these two dimensions coincide. Consequently, we prove that the Noetherian dimension of non-Noetherian Artinian serial modules over the rings of the title is 1.
MSC:
| 16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
| 16P20 | Artinian rings and modules (associative rings and algebras) |
| 16P40 | Noetherian rings and modules (associative rings and algebras) |
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