×

Zariski-finite modules. (English) Zbl 0980.13008

Let \(R\) be a commutative ring with identity, and let \(M\) be an \(R\)-module. For a submodule \(N\) of \(M\) let \((N:M)= \{r\in R\mid rM\subset N\}\). The submodule \(N\) is called prime if \((N:M)\) is a prime ideal of \(R\) and the \(R/ (N:M)\)-module \(M/N\) is torsion-free. In analogy to the spectrum of a ring one can define the set \(\text{Spec}(M)\) to be the collection of all prime submodules of \(M\). Let \(V(N)\), the variety of \(N\), be the collection of all prime submodules containing \(N\). The collection of all such varieties is denoted by \(\zeta(M)\). It can be shown that \(\zeta(M)\) is a semi-module over the semi-ring \(\zeta(R)\). This paper contains several results on generating sets of \(\zeta (M)\). For instance, if \(R\) is a domain and \(M\) is torsion-free and contains non-zero elements \(x,y\) such that \(Rx\cap Ry=0\), then the cardinality of every generating set of \(\zeta(M)\) has to be at least as large as the cardinality of \(R\).

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
54B35 Spectra in general topology
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] J. Dauns, Prime modules and one-sided ideals , in Ring theory and algebra III, Proc. 3rd Oklahoma Conf. (B.R. McDonald, ed.), Dekker, New York, 1980, 301-344. · Zbl 0457.16021
[2] A. El-Bast and P.F. Smith, Multiplication modules , Comm. Algebra 16 (1988), 755-759. · Zbl 0642.13002 · doi:10.1080/00927878808823601
[3] J.S. Golan, The theory of semirings with applications in mathematics and theoretical computer science , Pitman Monographs Surveys Pure Appl. Math., John Wiley & Sons, New York, 1992. · Zbl 0780.16036
[4] C.-P. Lu, Prime submodules of modules , Comm. Math. Univ. Sancti Pauli 33 (1984), 61-69. · Zbl 0575.13005
[5] ——–, \(M\)-radicals of submodules in modules , Math. Japon. 34 (1989), 211-219. · Zbl 0673.13007
[6] R.L. McCasland and M.E. Moore, Prime submodules , Comm. Algebra 20 (1992), 1803-1817. · Zbl 0776.13007 · doi:10.1080/00927879208824432
[7] R.L. McCasland, M.E. Moore and P.F. Smith, On the spectrum of a module over a commutative ring , Comm. Algebra 25 (1997), 79-105. · Zbl 0876.13002 · doi:10.1080/00927879708825840
[8] ——–, An introduction to Zariski spaces over Zariski topologies , Rocky Mountain J. Math., · Zbl 0939.13007 · doi:10.1216/rmjm/1181071721
[9] ——–, Modules with finitely generated spectra , Houston J. Math. 22 (1966), 457-471. · Zbl 0866.14001
[10] ——–, Generators for the semimodule of varieties of a free module over a commutative ring ,
[11] ——–, Modules with bounded spectra , · Zbl 0905.13003
[12] R.L. McCasland and P.F. Smith, Prime submodules of Noetherian modules , Rocky Mountain J. Math. 23 (1993), 1041-1062. · Zbl 0814.16017 · doi:10.1216/rmjm/1181072540
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.