×

Modules with bounded spectra. (English) Zbl 0905.13003

Summary: Let \(R\) be a commutative ring with identity and let \(M\) be an \(R\)-module. We examine the situation where for each prime ideal \({\mathfrak p}\) of \(R\) the set of all \({\mathfrak p}\)-prime submodules of \(M\) is finite. In case \(R\) is Noetherian and \(M\) is finitely generated, we prove that this condition is equivalent to there being a positive integer \(n\) such that for every prime ideal \({\mathfrak p}\) of \(R\), the number of \({\mathfrak p}\)-prime submodules of \(M\) is less than or equal to \(n\). We further show that in this case, there is at most one \({\mathfrak p}\)-prime submodule for all but finitely many prime ideals \({\mathfrak p}\) of \(R\).

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13E05 Commutative Noetherian rings and modules
13E15 Commutative rings and modules of finite generation or presentation; number of generators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Eisenbud D., Commutative algebra with a view toward algebraic geometry (1995) · Zbl 0819.13001
[2] Lu C.P., Comm. Math. Univ. Sancti Pauli 33 pp 61– (1984)
[3] DOI: 10.1080/00927879208824432 · Zbl 0776.13007 · doi:10.1080/00927879208824432
[4] McCasland R.L., Houston J 22 pp 457– (1996)
[5] DOI: 10.1216/rmjm/1181072540 · Zbl 0814.16017 · doi:10.1216/rmjm/1181072540
[6] McConnell J.C., Noncommutative Noetherian rings (1987) · Zbl 0644.16008
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.