Generalized SIP-modules

Yıl 2019, Cilt: 48 Sayı: 4, 1137 - 1145, 08.08.2019

Özgür Taşdemir Fatih Karabacak

Öz

We say an $R$-module $M$ has the generalized summand intersection property (briefly $GSIP$), if the intersection of any two direct summands is isomorphic to a direct summand. This is a generalization of SIP modules. In this note, the characterization of this property over rings and modules is investigated and some useful propositions obtained in SIP modules are generalized to GSIP modules.

Anahtar Kelimeler

SIP, GSIP, injective module, $V$-ring

Kaynakça

• [1] M. Alkan and A. Harmanci, On summand sum and summand intersection property of modules, Turk. J. Math. 26, 131–147, 2002.
• [2] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, New York, Springer-Verlag, 1974.
• [3] D.M. Arnold and J. Hausen, A characterization of modules with the summand intersection property, Comm. Algebra 18, 519–528, 1990.
• [4] G.F. Birkenmeier and A. Tercan, When some complement of a submodule is summand, Comm. Algebra, 35, 597–615, 2007.
• [5] G.F. Birkenmeier, B.J. Müller and S.T. Rizvi, Modules in which every fully invariant submodule is essential in a direct summand, Comm. Algebra, 30 (3), 1395–1415, 2002.
• [6] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending Modules, Pitman RN Mathematics 313, Longman, Harlow, 1994.
• [7] L. Fuchs, Infinite Abelian Groups I, Pure and Applied Mathematics, Academic Press, New York-London 1970.
• [8] A. Hamdouni, A. Harmanci and A.Ç. Özcan, Characterization of modules and rings by the summand intersection property and the summand sum property, JP J. Algebra Number Theory Appl. 5 (3), 469–490, 2005.
• [9] Hausen, J. Modules with the summand intersection property, Comm. Algebra 17 (1), 135-148, 1989.
• [10] I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1954.
• [11] I. Kaplansky, Commutative Rings, Univ. of Chicago Press, Chicago, 1974.
• [12] F. Karabacak and A. Tercan, On modules and matrix rings with SIP-extending, Taiwanese J. Math. 11 (4), 1037–1044, 2007.
• [13] S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series, 147 Cambridge, Cambridge Univ. Press, 1990.
• [14] P.F. Smith and A. Tercan, Continuous and quasi-continuous modules, Houston J. Math. 18 (3), 339–348, 1992.
• [15] P.F. Smith and A. Tercan, Generalizations of CS-modules, Comm. Algebra, 21, 1809– 1847, 1993.
• [16] D. Valcan, Injective modules with the direct summand intersection property, Sci. Bull. of Moldavian Academy of Sciences, Seria Mathematica, 31, 39–50, 1999.
• [17] G.V. Wilson, Modules with the summand intersection property, Comm. Algebra, 14, 21–38, 1986.
• [18] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.