Covering modules by proper submodules

Khare, Apoorva; Tikaradze, Akaki

doi:10.1080/00927872.2021.1959922

Cited by 5 publications

(7 citation statements)

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“…are rather straightforward, where S M is as above. The contribution in [9] was to show that the second inequality is in fact an equality for various classes of rings R and R-modules M . In this paper, (a) we show this equality holds for somewhat larger classes of finitely generated modules M ; and (b) we then show that the first inequality σ τ (M, R) ≤ σ(M, R) is also an equality, under reasonable technical assumptions.…”

Section: Overview Of Main Resultsmentioning

confidence: 99%

“…A well-known and well-studied question in group theory is "to find the minimum cardinal number σ(G) of proper subgroups of a given non-cyclic group G whose union covers G." This problem has a vast literature, much of which has been generously listed in the references of [9] (to which we direct any interested reader). One might ask an analogous question for vector spaces, and in general for modules.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we shall be concerning ourselves with the covering problem for finitely generated modules, and its relation to the version for vector spaces. A few instances where this problem or closely related variants have been addressed are [13], [6], [17] and most recently [9].…”

Section: Introductionmentioning

confidence: 99%

“…In [9], the authors studied the minimum (cardinal) number of proper submodules required to cover a module M over a unital commutative ring R, which we will denote in this paper by σ(M, R). The authors of [9] proved the equality σ(M, R) = min m∈S M |R/m| + 1, for several classes of rings R and R-modules, and also showed that S M = ∅ implies cyclicity in the cases considered. Here S M is the set of maximal ideals m of R for which the quotient module M/mM has dimension at least 2 as a vector space over the residue field R/m.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we show that the desired equality holds for finitely generated modules of finite dual Goldie dimension over arbitrary commutative rings with unity. One can then see that many classes of modules studied in [9] are special cases.…”

Section: Introductionmentioning

confidence: 99%

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The Zariski covering number for vector spaces and modules

Ghosh

2021

Preprint

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Given a K-vector space V , let σ(V, K) denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers V . Analogously, define σ(M, R) for a module M over a unital commutative ring R; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare-Tikaradze [Comm. Algebra, in press] showed for several classes of rings R and R-modules M that σ(M, R) = minm∈S M |R/m| + 1, where SM is the set of maximal ideals m such that dim R/m (M/mM ) ≥ 2. (That σ(M, R) ≤ minm∈S M |R/m| + 1 is straightforward.) Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension.We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose 'some' residue fields are infinite, which we call the Zariski covering number στ (M, R).To do so, we first define the "induced Zariski topology" τ on M , and now define στ (M, R) to be the smallest (cardinal) number of proper τ -closed subsets of M whose union covers M . We first show that our choice of topology implies that στ (M, R) ≤ σ(M, R), the covering number.We then show our next main result: στ (M, R) = minm∈S M |R/m| + 1, for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) m / ∈ SM whenever R/m is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number σ(M, R) of the aforementioned finitely generated modules.Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities στ (M, R) ≤ σ(M, R) ≤ κM := minm∈S M |R/m| + 1 again become equalities when the image of M under the continuous map q : M → m∈mSpec(R) M/mM (with appropriate Zariski-type topologies) is a κM -Baire subspace of the product space.

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Section: Overview Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%