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Let $R$ be a commutative ring and $M$ a non-zero $R$-module. We introduce the class of emph{pseudo strongly hollow submodules} (emph{PS-hollow submodules}, for short) of $M$. Inspired by the theory of modules with emph{secondary representations}, we investigate modules which can be written as emph{finite} sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of emph{minimal} PS-hollow strongly representations of modules over Artinian rings.
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Let $R$ be a commutative ring. We investigate $R$-modules which can be written as emph{finite} sums of {it {second}} $R$-submodules (we call them emph{second representable}). We provide sufficient conditions for an $R$-module $M$ to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (emph{main}) emph{second attached prime ideals} related to a module with such a presentation.
It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $widehat{R}$ of $R$ is of finite rank over the completion $widetilde{R}$ of $R$ in the $R$-topology. In this case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over one Krull-dimensional local Noetherian rings has finite Malcev rank. The preservation of Goldie dimension finiteness by localization is investigated too.
We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $igg 0$ implies that $M$ has finite projective dimension or finite injective dimension.
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multiple real parameters. The algebraic theory yields computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology indexed by arbitrary posets. The exposition focuses on the nature and ramifications of a suitable finiteness condition to replace the noetherian hypothesis. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both the theoretical and computational purposes: it guarantees finite primary decompositions, as well as various finite presentations and resolutions all related by a syzygy theorem, and the data structures thus produced are computable in addition to being interpretable. The tameness condition and its resulting theory are new even in the finitely generated discrete setting, where being tame is materially weaker than being noetherian.
It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t^{-1}]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a corollary.
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