
Algebra 2013
On Quasi Dense Submodules and Pure Envelopes of QTAG ModulesDOI: 10.1155/2013/873193 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
