English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We study Gorenstein dimension and grade of a module $M$ over a filtered ringwhose assosiated graded ring is a commutative Noetherian ring. An equality oran inequality between these invariants of a filtered module and its associatedgraded module is the most valuable property for an investigation of filteredrings. We prove an inequality G-dim$M\leq{G-dim gr}M$ and an equality ${\rmgrade}M={\rm grade gr}M$, whenever Gorenstein dimension of ${\rm gr}M$ isfinite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds anew viewpoint for studying filtered rings and modules. We apply these resultsto a filtered ring with a Cohen-Macaulay or Gorenstein associated graded ringand study a Cohen-Macaulay, perfect or holonomic module.

Cohen-Macaulay modules and holonomic modules over filtered rings