TL;DR
This paper explores the properties and classifications of Matlis reflexive modules over commutative rings, highlighting their structure as a Krull-Schmidt category and their relation to noetherian and artinian modules.
Contribution
It establishes that Matlis reflexive modules form a Krull-Schmidt category and characterizes their structure over noetherian rings, including classifications in specific cases.
Findings
Matlis reflexive modules form a Krull-Schmidt category.
Over noetherian rings, these modules lack infinite direct sums.
Classifications of Matlis reflexive modules are provided for small examples.
Abstract
Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. It is shown that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a characteristic feature of Matlis reflexivity. This leads to a discussion of objects that are extensions of artinian by noetherian objects. Classifications of Matlis reflexive modules are provided for some small examples.
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