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Keywords:
modular lattice; essential ideal; max-semicomplement; extending ideal; direct summand; exchangeable decomposition; ojective ideal
modular lattice; essential ideal; max-semicomplement; extending ideal; direct summand; exchangeable decomposition; ojective ideal
Summary:
The concept of an extending ideal in a modular lattice is introduced. A translation of module-theoretical concept of ojectivity (i.e. generalized relative injectivity) in the context of the lattice of ideals of a modular lattice is introduced. In a modular lattice satisfying a certain condition, a characterization is given for direct summands of an extending ideal to be mutually ojective. We define exchangeable decomposition and internal exchange property of an ideal in a modular lattice. It is shown that a finite decomposition of an extending ideal is exchangeable if and only if its summands are mutually ojective.
References:
[1] Akalan, E., Birkenmeier, G. F., Tercan, A.: Goldie extending modules. Commun. Algebra 37 (2009), 663-683; Corrigendum. 38 4747-4748 (2010); Corrigendum. 41 2005 (2013). DOI 10.1080/00927870802254843 | MR 2493810 | Zbl 1278.16005
[2] Birkenmeier, G. F., Müller, B. J., Rizvi, S. T.: Modules in which every fully invariant submodule is essential in a direct summand. Commun. Algebra 30 (2002), 1395-1415. DOI 10.1080/00927870209342387 | MR 1892606 | Zbl 1006.16010
[3] Grätzer, G.: General Lattice Theory. Birkhäuser Basel (1998). MR 1670580
[4] Grzeszczuk, P., Puczyłowski, E. R.: On finiteness conditions of modular lattices. Commun. Algebra 26 (1998), 2949-2957. DOI 10.1080/00927879808826319 | MR 1635878 | Zbl 0914.06003
[5] Grzeszczuk, P., Puczyłowski, E. R.: On Goldie and dual Goldie dimensions. J. Pure Appl. Algebra 31 (1984), 47-54. DOI 10.1016/0022-4049(84)90075-6 | MR 0738204 | Zbl 0528.16010
[6] Hanada, K., Kuratomi, Y., Oshiro, K.: On direct sums of extending modules and internal exchange property. J. Algebra 250 (2002), 115-133. DOI 10.1006/jabr.2001.9089 | MR 1898379 | Zbl 1007.16005
[7] Harmanci, A., Smith, P. F.: Finite direct sums of CS-modules. Houston J. Math. 19 (1993), 523-532. MR 1251607 | Zbl 0802.16006
[8] Kamal, M. A., Müller, B. J.: Extending modules over commutative domains. Osaka J. Math. 25 (1988), 531-538. MR 0969016 | Zbl 0715.13006
[9] Kamal, M. A., Müller, B. J.: The structure of extending modules over Noetherian rings. Osaka J. Math. 25 (1988), 539-551. MR 0969017 | Zbl 0715.16005
[10] Kamal, M. A., Müller, B. J.: Torsion free extending modules. Osaka J. Math. 25 (1988), 825-832. MR 0983803 | Zbl 0703.13010
[11] Lam, T. Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics 189 Springer, New York (1999). DOI 10.1007/978-1-4612-0525-8 | MR 1653294
[12] Mohamed, S. H., Müller, B. J.: Ojective modules. Commun. Algebra 30 (2002), 1817-1827. DOI 10.1081/AGB-120013218 | MR 1894046 | Zbl 0998.16005
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