The Zariski topology on the graded classical prime spectrum of a graded module over a graded commutative ring. (English) Zbl 1474.13002

Summary: Let \(G\) be a group with identity \(e\). Let \(R\) be a \(G\)-graded commutative ring and \(M\) a graded \(R\)-module. A proper graded submodule \(N\) of \(M\) is called a graded classical prime if whenever \(r,s\in h(R)\) and \(m\in h(M)\) with \(rsm\in N\), then either \(rm\in N\) or \(sm\in N\). The graded classical prime spectrum \(\mathrm{Cl.Spec}^g(M)\) is defined to be the set of all graded classical prime submodules of \(M\). In this paper, we introduce and study a topology on \(\mathrm{Cl.Spec}^g(M)\), which generalizes the Zariski topology of graded ring \(R\) to graded module \(M\), called Zariski topology of \(M\), and investigate several properties of the topology.


13A02 Graded rings
Full Text: Link Link


[1] R. Abu-Dawwas, K. Al-Zoubi,On graded weakly classical prime submodules, Iran. J. Math. Sci. Inform., (in press). · Zbl 1374.13002
[2] K. Al-Zoubi, R. Abu-Dawwas,On graded quasi-prime submodules, Kyungpook Math. J.,55 (2) (2015), 259-266. · Zbl 1325.13006
[3] K. Al-Zoubi, M. Al-Dolat,On graded classical primary submodules, Adv. Pure Appl. Math., 7(2) (2016), 93-96. · Zbl 1335.13003
[4] K. Al-Zoubi, M. Jaradat, R. Abu-Dawwas,On graded classical prime and graded prime submodules, Bull. Iranian Math. Soc.,41(1) (2015), 217-225. · Zbl 1335.13002
[5] K. Al-Zoubi, F. Qarqaz,An intersection condition for graded prime submodules in Grmultiplication modules, Math. Reports, (in press). · Zbl 07002433
[6] A. Abbasi, D. Hassanzadeh,Modules and spectral spaces, Comm. Algebra,40(2012), 4111- 4129. · Zbl 1262.13015
[7] H. Ansari-Toroghy, R. Ovlyaee-Sarmazdeh,On the prime spectrum of a module and Zariski topologies, Comm. Algebra,38(2010), 4461-4475. · Zbl 1206.13016
[8] H. Ansari-Toroghy, S.S. Pourmortazavi,On the prime spectrum of modules, Miskolc Math. Notes,16(2) (2015), 1233-1242. · Zbl 1349.13027
[9] S.E. Atani,On graded prime submodules, Chiang Mai. J. Sci.,33(1) (2006), 3-7. · Zbl 1099.13001
[10] S.E. Atani, F. Farzalipour,On graded secondary modules, Turk. J. Math.,31(2007), 371-378. · Zbl 1132.13001
[11] S.E. Atani, F.E.K. Saraei,Graded modules which satisfy the Gr-Radical formula, Thai J. Math.,8(1) (2010), 161-170. · Zbl 1214.13001
[12] A.Y. Darani,Topologies onSpecg(M), Bul. Acad. Stiinte Repub, Mold. Mat.,3(67)(2011), 45-53.
[13] A.Y. Darani, S. Motmaen,Zariski topology on the spectrum of graded classical prime submodules, Appl. Gen. Topol.,14(2) (2013), 159-169 · Zbl 1309.13003
[14] R. Hazrat,Graded Rings and Graded Grothendieck Groups, Cambridge University Press, Cambridge, 2016. · Zbl 1390.13001
[15] M. Hochster,Prime ideal structure in commutative rings, Trans. Amer. Math. Soc.,137 (1969), 43-60. · Zbl 0184.29401
[16] C.P. Lu,Modules with Noetherian spectrum, Comm. Algebra,38(2010), 807-828. · Zbl 1189.13012
[17] C.P. Lu,The Zariski topology on the prime spectrum of a module, Houston J. Math.,25(3) (1999), 417-432. · Zbl 0979.13005
[18] J.R. Munkres,Topology; A first course, Prentice-Hall, Inc. Eaglewood Cliffs, New Jersey, 1975.
[19] R.L. McCasland, M. E. Moore, P. F. Smith,On the spectrum of a module over a commutative ring, Comm. Algebra,25(1997), 79-103. · Zbl 0876.13002
[20] C. Nastasescu, V.F. Oystaeyen,Methods of Graded Rings, LNM 1836. Berlin-Heidelberg: Springer-Verlag, 2004. · Zbl 1043.16017
[21] C. Nastasescu, F. Van Oystaeyen, Graded Ring Theory, Mathematical Library 28, North Holand, Amsterdam, 1982. · Zbl 0494.16001
[22] N.A. Ozkirisci , K.H. Oral, U. Tekir,Graded prime spectrum of a graded module, Iran. J. Sci. Technol.,37A3(2013), 411-420.
[23] K.H. Oral, U. Tekir, A. G. Agargun,On graded prime and primary submodules, Turk. J. Math.,35(2011), 159-167. · Zbl 1279.13004
[24] R.N. Uregen, U. Tekir, K.H. Oral,On the union of graded prime ideals, Open Phys.,14(2016), 114-118.
[25] M. Refai,On properties of G-spec(R), Sci. Math. Jpn.,4(2001), 491-495. · Zbl 0998.13002
[26] M. Refai, K. Al-Zoubi,On graded primary ideals, Turk. J. Math.,28(2004), 217-229. · Zbl 1077.13001
[27] H.A.Tavallaee, M. Zolfaghari,Graded weakly semiprime submodules of graded multiplication modules, Lobachevskii J. Math.,34(1) (2013), 61-67 · Zbl 1274.13019
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.