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--dense in , if for every -pure submodule , containing ,   is -divisible.

Lemma 2. If for some integer , then there exists a proper submodule of containing and a bounded submodule of such that (i); (ii); (iii).

Proof. The socle of can be expressed as the direct sum of where and . Now there exists a -pure submodule of such that . Also is -pure in and . Now Soc which is contained in ; therefore is an absolute direct summand of [4]. Thus there exists a submodule of such that and ; therefore and .

Proposition 3. A submodule of a module is quasi--dense in if and only if for all integers .

Proof. Let be the least positive integer such that . By Lemma 2, there exists a proper submodule containing and a bounded submodule of such that . This contradiction proves that , for all integers .
For the converse, consider an -pure submodule . If is not -divisible, then there exist submodules and of such that , where is bounded. If and , then . Since is -pure in , we have . On repeating the process, after a finite number of steps, we get and , which is a contradiction. Hence, is -divisible and is quasi--dense in .

Remark 4. For a submodule of a module , the following are equivalent: (i) is quasi--dense in ; (ii); (iii)for every -pure submodule containing , is -divisible.

Lemma 5. If is an -divisible submodule of a module and is a -high submodule of , then is -pure in .

Proof. For an -divisible submodule  , -high submodules are -neat [5]. Therefore, . Assume that for all . Let . Now there exists such that . If such that , then for some and . Now there exists such that and . By induction and is -pure in .

Theorem 6. Let be a proper submodule of . Then is not semi--pure if and only if the following hold: (i) is quasi--dense in ;(ii)there exists such that .

Proof. By Lemma 2 and Proposition 3,   is satisfied. Suppose , for all . If there exists a nonnegative integer such that then we have Let be the -neat envelope [6] of in . Then is -pure in . Since is quasi--dense in ,   is -divisible. Thus, there exists a submodule of such that and Now is -pure in , hence and . Therefore, we may assume that there exists a chain of integers such that Then we may choose such that for every and we have . Let be an -high submodule of such that . There exists a proper unbounded basic submodule of and a submodule , containing such that . Now is -pure in which is a contradiction proving (ii).
Conversely, suppose is an -pure submodule of containing . By (i),   is -divisible. Since and is -pure in , we have .

Now we should mention the following notations used by Khan and Zubair [7]: We prove the following.

Proposition 7. Let be an -pure submodule of a QTAG module . Then , for all .

Proof. Since and , there exists a homomorphism such that . If and , then there exist and such that and . Moreover if , , , then is contained in and there exists such that , where and for some and , respectively. Since and , we have . Also is one to one by definition; hence is an isomorphism.

Theorem 8. Let be a semi--pure submodule in having an -pure hull. Then there exists an integer such that , for .

Proof. Let be an -pure hull of in . By Theorem 6, there exists a nonnegative integer such that and for all ,
Therefore , for all . Hence by Proposition 7, for all .

Proposition 9. Let be an -pure hull of a semi--pure submodule of and a nonnegative integer. Then and if and only if for  all and .

Proof. Since for all , by Proposition 7,  . If , then and . This contradiction implies that .
For the converse, consider the integer such that and . If , then which is a contradiction. If , then Again it is a contradiction, hence .

modules are defined in [3], and we can observe the following.

Remark 10. If is -pure in , then by Proposition 7, and implies that for all . In other words, is a module in .

Proposition 11. Let be an -neat submodule of . Then is -pure in if and only if is a module in .

Proof. Suppose . Consider such that there exists and . This implies the existence of such that and . Now there exists , such that , where . As , we have for some and . Since , there exists such that and is -pure in . The converse is trivial.

Proposition 12. A submodule is a module if and only if for all .

Proof. Suppose is a module in . Therefore . We will prove the result by induction. Assume that for all . Since , we have The converse is trivial.

Following results are the immediate consequences of the previous discussion.

Remark 13. (i) is a module in if and only if for all .
(ii) If for all , then is a module in .

3. -Pure Envelopes

For a proper submodule , it is not always possible for to have an -pure hull in . We study the situation when there is a proper -pure submodule of , but no proper direct summand of contains . This motivates us to define -pure envelopes like -neat envelope defined earlier [6]. We find that the -Kaplansky invariants are same for all -pure envelopes.

Definition 14. Let be a submodule of a QTAG module . An -pure submodule is an -pure envelope of if has no proper direct summand containing .

Proposition 15. In a module , an -pure submodule is an -pure envelope of if and only if contains no -divisible summand disjoint from and for any , no uniserial summand of decomposition length disjoint from .

Proof. If is not an -pure envelope of , then with . If is not -reduced, then it contains an -divisible summand disjoint from , and if is -reduced, then it contains a uniserial summand such that for some . Without loss of generality, we may assume that . Now implying that . Therefore, and .
Conversely, if is -divisible submodule of with , then is a summand of ; therefore cannot be an -pure envelope of . If , it is a uniserial summand such that and , then such that . This can be chosen to be any submodule of which is maximal with respect to the properties of containing and disjoint from . This can not be the -pure envelope of .

Let be a uniserial summand of such that . If such that , then if and only if . Since , this implies that . This enables us to prove the following.

Proposition 16. An -pure submodule containing the submodule of a separable module is an -pure envelope if and only if , for all .

Proof. Let be a semi--pure submodule of a separable module and the -pure hull of . Now we have Soc . This implies that for every . This is true for -pure envelopes, and if is separable, this is sufficient also.

Remark 17. Since for all , if and only if Soc for every , an -pure submodule is an -pure envelope of if and only if , for every .

Remark 18. Since the union of a chain of -pure envelopes of may not contain any -divisible direct summand disjoint from or a uniserial summand of decomposable length disjoint from , every -pure envelope of is contained in a maximal -pure envelope of .

Now we investigate -pure envelopes of containing other -pure envelopes of .

Theorem 19. Let be a submodule of a separable module and , -pure submodules of containing . Then (i)for every , the natural embedding induces a monomorphism (ii)the map is onto for every if and only if is an -dense submodule of .