On spectrum of comultiplication modules. (English) Zbl 1484.13023
In this article, comultiplication modules in terms of dual Zariski topology are characterized. Some connections between subspaces of a complement dual Zariski topology and submodules of a comultiplication module are studied. Then, some useful relations between open subsets of a complement dual Zariski topology and second radical submodules related to these are obtained. Finally, direct connections among rings, comultiplication modules, the complement dual Zariski topology and the dual Zariski topology are established.
Reviewer: Chen Sheng (Harbin)
MSC:
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
| 13C13 | Other special types of modules and ideals in commutative rings |
| 13C99 | Theory of modules and ideals in commutative rings |
| 13E99 | Chain conditions, finiteness conditions in commutative ring theory |
| 54B99 | Basic constructions in general topology |
Keywords:
comultiplication modules; dual Zariski topology; Noetherian spectrum; second submodules; second radicalReferences:
| [1] | Ansari-Toroghy, H.; Keyvani, S., On the maximal spectrum of a module and Zariski topology, Bull. Malays. Math. Sci. Soc, 38, 1, 303-316 (2015) · Zbl 1308.13031 · doi:10.1007/s40840-014-0020-1 |
| [2] | Ansari-Toroghy, H.; Keyvani, S.; Farshadifar, F., The Zariski topology on the second spectrum of a module II, Bull. Malays. Math. Sci. Soc, 39, 3, 1089-1103 (2016) · Zbl 1348.13019 · doi:10.1007/s40840-015-0225-y |
| [3] | Ansari-Toroghy, H.; Ovlyaee-Sarmazdeh, R., On the prime spectrum of a module and Zariski topologies, Commun. Algebra, 38, 12, 4461-4475 (2010) · Zbl 1206.13016 · doi:10.1080/00927870903386510 |
| [4] | Atiyah, M. F.; MacDonald, I. G., Introduction to Commutative Algebra (1969), London, UK: Addison-Wesley, London, UK · Zbl 0175.03601 |
| [5] | Çeken, S.; Alkan, M., On the second spectrum and the second classical Zariski topology of a module, J. Algebra Appl, 14, 10, 1550150 (2015) · Zbl 1333.16040 · doi:10.1142/S0219498815501509 |
| [6] | Çeken, S.; Alkan, M., On the weakly second spectrum of a module, Filomat., 30, 4, 1013-1020 (2016) · Zbl 1474.16001 · doi:10.2298/FIL1604013C |
| [7] | Çeken, S.; Alkan, M., Second spectrum of modules and spectral spaces, Bull. Malays. Math. Sci. Soc, 42, 1, 153-169 (2019) · Zbl 1408.13021 · doi:10.1007/s40840-017-0473-0 |
| [8] | Çeken, S.; Alkan, M., Dual of Zariski topology for modules, AIP Conf. Proc., 1389, :357-360 (2011) |
| [9] | Kunz, E., Introduction to Commutative Algebra and Algebraic Geometry (1985), Basel, Switzerland: Birkhauser, Basel, Switzerland · Zbl 0563.13001 |
| [10] | Lu, C. P., Modules with Noetherian spectrum, Commun. Algebra, 38, 3, 807-828 (2010) · Zbl 1189.13012 · doi:10.1080/00927870802578050 |
| [11] | McCasland, R. L.; Moore, M. E.; Smith, P. F., An introduction to Zariski spaces over Zariski topologies, Rocky Mountain J. Math, 28, 4, 1357-1369 (1998) · Zbl 0939.13007 · doi:10.1216/rmjm/1181071721 |
| [12] | Öneş, O., Dual Zariski topology on Comultiplication Modules, Sakarya Univ. J. Sci., 23, 3, 390-395 (2019) |
| [13] | Öneş, O.; Alkan, M., The structure of some topologies on a module, AIP Conf. Proc., 1863, 300010(1)-300010(4) (2017) |
| [14] | Öneş, O.; Alkan, M., A note on graded ring with prime spectrum, Adv. Stud. Cont. Math, 28, 4, 625-634 (2018) · Zbl 1414.13003 |
| [15] | Öneş, O.; Alkan, M., Multiplication modules with prime spectrum, Turk. J. Math, 43, 4, 2000-2009 (2019) · Zbl 1429.13011 · doi:10.3906/mat-1803-54 |
| [16] | Öneş, O.; Alkan, M., Zariski subspaces topologies on ideals, HJMS, 48, 6, 1667-1674 (2019) · Zbl 1499.13049 · doi:10.15672/HJMS.2018.597 |
| [17] | Öneş, O.; Alkan, M., Dual Zariski Topology Related to Ideals of a Ring (2019) |
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.
