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Algebra  2013

# On Quasi- -Dense Submodules and -Pure Envelopes of QTAG Modules

Abstract:

A module over an associative ring with unity is a QTAG module if every finitely generated submodule of any homomorphic image of is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which -pure submodules and high submodules are significant. A submodule is quasi- -dense in if is -divisible, for every -pure submodule of containing Here we study these submodules and obtain some interesting results. Motivated by -neat envelope, we also define -pure envelope of a submodule as the -pure submodule if has no direct summand containing We find that -pure envelopes of have isomorphic basic submodules, and if is the direct sum of uniserial modules, then all -pure envelopes of are isomorphic. 1. Introduction All the rings considered here are associative with unity, and right modules are unital modules. An element is uniform, if is a nonzero uniform (hence uniserial) module and for any -module with a unique decomposition series, denotes its decomposition length. For a uniform element , , and are the exponent and height of in , respectively. denotes the submodule of generated by the elements of height at least , and is the submodule of generated by the elements of exponent at most . ？ is -divisible if , and it is -reduced if it does not contain any -divisible submodule. In other words, it is free from the elements of infinite height. The modules , form a neighbourhood system of zero giving rise to -topology. The closure of a submodule is defined as , and it is closed with respect to -topology if . A submodule of is -pure in if , for every integer . For a limit ordinal , , for all ordinals , and it is -pure in if for all ordinals . A module is summable if , where is the set of all elements of which are not in , where is the length of . A submodule is nice [1, Definition 2.3] in , if for all ordinals ; that is, every coset of modulo may be represented by an element of the same height. The cardinality of the minimal generating set of is denoted by . For all ordinals , is the - invariant of and it is equal to . For a module , there is a chain of submodules , for some ordinal . , where is the submodule of . Singh [2] proved that the results which hold for TAG modules also hold good for modules. 2. Quasi- -Dense Submodules In [3], we studied semi- -pure submodules which are not -pure but contained in -pure submodules. Now we investigate the submodules such that is -divisible for every -pure submodule , containing . These modules are called quasi- -dense submodules. We start with the following. Definition 1. A submodule of is quasi-

References

 [1] M. Z. Khan, “On basic submodules,” Tamkang Journal of Mathematics, vol. 10, no. 1, pp. 24–29, 1979. [2] S. Singh, “Some decomposition theorems in abelian groups and their generalizations,” in Ring Theory, Proc. Ohio Univ. Cong., pp. 183–189, Marcel Dekker, New York, NY, USA, 1977. [3] A. Mehdi and F. Sikander, “Some characterizations of submodules of QTAG-modules,” Scientia. Series A, vol. 18, pp. 39–46, 2009. [4] F. Mehdi and A. Mehdi, “ -high submodules and -topology,” South East Asian Journal of Mathematics and Mathematical Sciences, vol. 1, no. 1, pp. 83–88, 2002. [5] M. Z. Khan, “ -divisible and basic submodules,” Tamkang Journal of Mathematics, vol. 10, no. 2, pp. 197–203, 1979. [6] A. Mehdi and M. Z. Khan, “On -neat envelopes and basic submodules,” Tamkang Journal of Mathematics, vol. 16, no. 2, pp. 71–76, 1985. [7] M. Z. Khan and A. Zubair, “On quasi -pure submodules of QTAG-modules,” International Journal of Mathematics and Mathematical Sciences, vol. 24, no. 7, pp. 493–499, 2000. [8] A. Mehdi and M. Z. Khan, “On closed modules,” Kyungpook Mathematical Journal, vol. 24, no. 1, pp. 45–50, 1984.

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