On the dual notion of prime submodules. (English) Zbl 1294.13012
Summary: Let \(R\) be a commutative ring and \(M\) an \(R\)-module. In this paper, we study the dual notion of prime submodules (that is, second submodules of \(M\)) and investigate the conditions under which the number of maximal second submodules of \(M\) is finite. Furthermore, we introduce the concept of coisolated submodules of \(M\) and obtain some related characterizations.
MSC:
| 13C13 | Other special types of modules and ideals in commutative rings |
Keywords:
second submodules; completely irreducible submodules; coisolated submodules; prime submodulesReferences:
| [1] | H. Ansari-Toroghy and F. Farshadifar, Taiwanese J. Math. 11(4), 1189 (2007). |
| [2] | H. Ansari-Toroghy and F. Farshadifar, CMU. J. Nat. Sci. 8(1), 105 (2009). |
| [3] | H. Ansari-Toroghy, F. Farshadifar, Fully idempotent and coidempotent modules, Bull. Iran. Math. Soc. (to appear) . · Zbl 1311.13010 |
| [4] | L. Fuchs, W. Heinzer and B. Olberding, Abelian Groups, Rings, Modules, and Homological Algebra, Lect. Notes Pure Appl. Math 249 (Chapman & Hall/CRC, Boca Raton, FL, 2006) pp. 121-145. |
| [5] | H. Matsumara , Commutative Ring Theory ( Cambridge University Press , Cambridge , 1986 ) . |
| [6] | R. L. McCasland and P. F. Smith, Rocky Mountain J. Math. 23(3), 1041 (1993), DOI: 10.1216/rmjm/1181072540. |
| [7] | R. L. McCasland and P. F. Smith, Comm. Algebra 34, 2977 (2006), DOI: 10.1080/00927870600639773. |
| [8] | R. L. McCasland and P. F. Smith, Int. Electron. J. Algebra 4, 159 (2008). |
| [9] | R. Wisbauer , Foundations of Module and Ring Theory ( Gordon & Breach , Philadelphia, PA , 1991 ) . · Zbl 0746.16001 |
| [10] | S. Yassemi, Arch. Math. (Brno) 37, 273 (2001). |
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