×
Behboodi, Mahmood; Sabzevari, Masoud

Modules satisfying the prime and maximal radical conditions. (English) Zbl 1309.13012

J. Commut. Algebra 6, No. 4, 505-523 (2014).
Given a commutative ring \(R\), an \(R\)-module \(M\) is \(\mathbb{P}\)-radical whenever \(M\) satisfies the equality \((\sqrt[p]{\mathcal{P}M}:M)=\sqrt{ \mathcal{P}}\), for every prime ideal \(\mathcal{P}\supseteq\) Ann\((\mathcal{P}M)\), where \(\sqrt[p]{\mathcal{P}M}\) is the intersection of all prime submodules of \(M\) containing \(\mathcal{P}M\). Among other results, the authors show that the class of \(\mathbb{P}\)-radical modules is wider than the class of primeful modules (introduced by C.-P. Lu [Houston J. Math. 33, No. 1, 125–143 (2007; Zbl 1114.13010)]). Also, they prove that any projective module over a Noetherian ring is \(\mathbb{P}\)-radical. This fact also holds for any arbitrary module over an Artinian ring. On the other hand, they call an \(R\)-module \(M\) by \(\mathbb{M}\)-radical if \((\sqrt[p]{\mathcal{M}M}: M)=\mathcal{M}\), for every maximal ideal \(\mathcal{M}\) containing Ann(\(M\)).
They also show that the conditions \(\mathbb{P}\)-radical and \(\mathbb{M}\)-radical are equivalent for all \(R\)-modules if and only if \(R\) is a Hilbert ring. Also, two conditions primeful and \(\mathbb{M}\)-radical are equivalent for all \(R\)-modules if and only if dim\(({R})=0\). Finally, they anticipate that the results of this paper will be applied in a subsequent work to construct a structure sheaf on the spectrum of \(\mathbb{P}\)-radical modules.
Reviewer: Marco Fontana (Roma)

Cited in 1 Document

MSC:

13C13 Other special types of modules and ideals in commutative rings
13A99 General commutative ring theory
13C99 Theory of modules and ideals in commutative rings
14A25 Elementary questions in algebraic geometry

Keywords:

prime submodule; Zariski topology; sheaf of rings; sheaf of modules; radical ideal

Citations:

Zbl 1114.13010
Cite Review PDF
Full Text: DOI arXiv Euclid

References:

[1] M. Aghasi, M. Behboodi and M. Sabzevari, A structure sheaf on the spectrum of prime radical modules , J. Comm. Alg.,
[2] M.F. Atiyah and I. Macdonald, Introduction to commutative algebra , Addison-Wesley Pub. Co., 1969.
[3] A. Barnard, Multiplication modules , J. Alg. 71 (1981), 174-178. · Zbl 0468.13011 · doi:10.1016/0021-8693(81)90112-5
[4] M. Behboodi, A generalization of the classical Krull dimension for modules , J. Alg. 305 (2006), 1128-1148. · Zbl 1114.16020 · doi:10.1016/j.jalgebra.2006.04.010
[5] —-, A generalization of Bear’s lower nilradical for modules , J. Alg. Appl. 6 (2007), 337-353. · Zbl 1119.16020 · doi:10.1142/S0219498807002284
[6] —-, On the prime radical and Baer’s lower nilradical of modules , Acta Math. Hung. 122 (2009), 293-306. · Zbl 1188.16014 · doi:10.1007/s10474-008-8028-3
[7] M. Behboodi, O.A.S. Karamzadeh and H. Koohy, Modules whose certain submodules are prime , Vietnam J. Math. 32 (2004), 303-317. · Zbl 1081.16008
[8] B. Cortzen and L.W. Small, Finite extensions of rings , Proc. Amer. Math. Soc. 103 (1988), 1058-1062. · Zbl 0654.16012 · doi:10.2307/2047085
[9] J. Dauns, Prime modules , J. reine angew. Math. 298 (1978), 156-181. · Zbl 0365.16002 · doi:10.1515/crll.1978.298.156
[10] D. Eisenbud, Commutative algebra with a view toward algebraic geometry , Springer-Verlag, New York, 1995. · Zbl 0819.13001
[11] Z.A. El-Bast and P.F. Smith, Multiplication modules , Comm. Alg. 16 (1988), 755-779. · Zbl 0642.13002 · doi:10.1080/00927878808823601
[12] E.H. Feller and E.W. Swokowski, Prime modules , Canad. J. Math. 17 (1965), 1041-1052. · Zbl 0138.26603 · doi:10.4153/CJM-1965-099-5
[13] O. Goldman, Hilbert rings and the Hilbert Nullstellensatz , Math. Z. 54 (1951), 136-140. · Zbl 0042.26401 · doi:10.1007/BF01179855
[14] R. Hamsher, Commutative rings over which every module has a maximal submodule , Proc. Amer. Math. Soc. 18 (1967), 1133-1137. · Zbl 0156.04303 · doi:10.2307/2035815
[15] R. Hartshorne, Algebraic geometry , Springer, New York, 1977. · Zbl 0367.14001
[16] C.P. Lu, Prime submodules of modules , Comm. Math. Univ. St. Paul 33 (1984), 61-69. · Zbl 0575.13005
[17] —-, Spectra of modules , Comm. Alg. 23 (1995), 3741-3752. · Zbl 0853.13011 · doi:10.1080/00927879508825430
[18] —-, The Zariski topology on the prime spectrum of a module , Houston J. Math. 25 (1999), 417-433. · Zbl 0979.13005
[19] —-, A module whose prime spectrum has the suejective natural map , Houston J. Math. 33 (2007), 125-143. · Zbl 1114.13010
[20] —-, \(M\)-radical of submodules in modules , Math. Japon. 34 (1989), 211-219. · Zbl 0673.13007
[21] S.H. Man, One dimensional domains which satisfy the radical formula are Dedekind domains , Arch. Math. 66 (1996), 276-279. · Zbl 0854.13010 · doi:10.1007/BF01207828
[22] R.L. McCasland and M.E. Moore, Prime submodules , Comm. Alg. 20 (1992), 1803-1817. · Zbl 0776.13007 · doi:10.1080/00927879208824432
[23] R.L. McCasland, M.E. Moore and P.F. Smith, On the spectrum of a module over a commutative ring , Comm. Alg. 25 (1997), 79-103. · Zbl 0876.13002 · doi:10.1080/00927879708825840
[24] 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
[25] P.F. Smith and D.P. Yilmaz, Radicals of submodules of free modules , Comm. Alg. 27 (1999), 2253-2266. · Zbl 0944.13001 · doi:10.1080/00927879908826563
[26] U. Tekir, On the sheaf of modules , Comm. Alg. 33 (2005), 2557-2562. · Zbl 1082.13500 · doi:10.1081/AGB-200065136
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.