On Totally Projective QTAG-modules Characterized by its Submodules

A QTAG-module M is called almost totally projective if it has a weak nice system. Here we show that the isotype submodules of a totally projective module which are almost totally projective are precisely those that are separable. From this characterization it follows that every balanced submodule of a totally projective module is almost totally projective. Finally, in some special cases we settle the question of whether a direct summand of an almost totally projective module is again almost totally projective.


Introduction and background material
Following [8], a unital module M R is called QT AG-module if it satisfies the following condition: (I) Every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules.
Let all rings discussed here be associative with unity (1 = 0) and modules are unital QT AG-modules.A module in which the lattice of its submodule is totally ordered is called a serial module; in addition, if it has finite composition length it is called a uniserial module.An element x ∈ M is uniform, if xR is a non-zero uniform (hence uniserial) module and for any R-module M with a unique decomposition series, d(M ) denotes its decomposition length.
H k (M ) and it is h-reduced if it does not contain any h-divisible submodule.In other words it is free from the elements of infinite height.
For an ordinal σ, a submodule N of M is said to be σ-pure, if H β (M ) ∩ N = H β (N ) for all β ≤ σ and a submodule N of M is said to be isotype in M , if it is σ-pure for every ordinal σ [7].A submodule N ⊂ M is nice [5] in M, if H σ (M/N ) = (H σ (M ) + N )/N for all ordinals σ, i.e. every coset of M modulo N may be represented by an element of the same height.If N is both isotype and nice, then N is called a balanced submodule of M .
A family N of nice submodules of M is called a nice system [6] (iii + ) given any N ∈ N and any countable subset X of M, there exists K ∈ N containing N ∪ X, such that K/N is countably generated.
An h-reduced QT AG-module M is totally projective if it has a nice system and direct sums and direct summands of totally projective modules are also totally projective [4].
In this paper, we focus on the class of isotype separable submodules of totally projective modules.The object of our study includes, the well known class of balanced submodules of totally projective modules.It is well known that a balanced submodule of a totally projective module need not be totally projective.Nevertheless, we show that the balanced submodules of totally projective modules are almost totally projective.It should be no surprise that this is done in the context of nice system.In addition to proving that every balanced submodule of a totally projective module is almost totally projective, we are able to characterize the isotype submodules of totally projective modules that are almost totally projective.This class will be seen to coincide with the class of isotype separable submodules of totally projective modules.Finally, as an application of this characterization and its method of proof, we obtain results concerning direct summands of almost totally projective modules.Our notations and terminology generally agree with those in [1] and [2].

* -bases and intersection closure
A QT AG-module M is almost totally projective if it has a collection N of nice submodules such that (i) 0 ∈ N, (ii) N is closed with respect to unions of ascending chains, and (iii) every countably generated submodule of M is contained in a countably generated submodule from N. Call a collection N of nice submodules of a QT AG-module M which satisfies conditions (i), (ii) and (iii) a weak nice system for M .It will be convenient to consider those almost totally projective modules that On Totally Projective QT AG-modules 79 have the following additional property.We say that an almost totally projective module is intersection closed if it has a weak nice system which is closed under the formation of arbitrary intersections.Our first objective in this section is to demonstrate that every totally projective module is intersection closed.This is done somewhat indirectly by exploiting the theory of * -bases as introduced in [3].
Let M be a QT AG-module.For each ordinal σ, let B σ be a set of representatives of the nonzero cosets of H σ (M ) mod H σ+1 (M ); in other words, B σ contains exactly one element from each of the nonzero cosets of H σ+1 (M ) in H σ (M ).If each element x in M can be expressed as The consequences of the results of [3] that concern in this paper are that every secure submodule is nice and that the collection N of all secure submodules of M constitutes a weak nice system for M .A novelty of our approach in this paper is based on the observation that an arbitrary intersection of secure submodules is again secure.This follows easily from the uniqueness of the representations for the elements of M as described above.Thus, every QT AG-module with a * -basis is intersection closed.In [3] it is also shown that every totally projective module has a * -basis.Therefore, we have the following.Proposition 2.1.If M is a totally projective module, then M is intersection closed.
We next establish a property of intersection closed modules that will use in the proof of Theorem 3.1.In order to state this result, we need some further notation and terminology.Suppose that M is a QT AG-module.If x ∈ M we write H M (x) for the height of x in M .Thus, Let M be a QT AG-module.Two submodules P and Q of M are compatible, written P Q, if for each pair (p, q) ∈ P × Q there exists r It is easily seen that compatibility is a symmetric relation, and is inductive in the sense that if is an ascending chain of submodules of M with P α Q for all α, then ( Lemma 2.1.Suppose K is a submodule of an intersection closed QT AG-module M .Let N be a weak nice system for M which is closed under arbitrary intersections and assume that is an ascending chain of submodules of M with P α ∈ N for all α.Then, there exists a submodule Q ∈ N such that the following properties hold. (i) Consequently, if P α K for all α, then Q α K for all α.Therefore, (ii) follows since compatibility with K is an inductive property.✷

A characterization of separable isotype totally projective modules
We begin with the following elementary lemma.
and the result follows.✷ The notion of separability has an important role in the study of QT AG-modules.The authors claim that the same should be true for submodules of the QT AGmodules, and in fact the remaining part of this paper substantially supports this claim.We now define separable submodule in a slightly different way from that in [2].
A submodule N of a QT AG-module M is a separable submodule if for each x ∈ M there is a corresponding countably generated submodule K of N such that sup{H M (x + y) : y ∈ N } = sup{H M (x + z) : z ∈ K}.
On Totally Projective QT AG-modules 81 There are two crucial properties of separability.
(A) An almost totally projective module is a separable submodule of any QT AG-module in which it appears as an isotype submodule.
(B) Suppose N is a separable submodule of a QT AG-module M .If S is a countably generated submodule of M , there exists a countably generated submodule T of M such that S ⊆ T and T N .
Our next result characterizes those isotype submodules of a totally projective module that are almost totally projective.Theorem 3.1.Suppose K is an isotype submodule of a totally projective module M .Then K is almost totally projective if and only if K is separable in M .
Proof: If K is almost totally projective, then K is separable in M by property (A).Conversely, assume that K is separable in M .By Proposition 2.1, M is intersection closed and so has a weak nice system N which is closed under intersection.Set We claim that N K is a weak nice system for K.
By Lemma 3.1, N K consists of nice submodules of K. Also, it is clear that 0 ∈ N K .Moreover, that N K is closed with respect to ascending unions follows immediately from Lemma 2.1.Therefore, to establish the claim, and thereby complete the proof, it suffices to show that every countably generated submodule S of K is contained in a countably generated submodule from N K .Since S is a countably generated submodule of M , there exists a countably generated submodule T 0 ∈ N such that S ⊆ T 0 .Next, K is separable in M and T 0 is countably generated, so it follows from property (B) that there exists a countably generated submodule S 0 of M such that T 0 ⊆ S 0 and S 0 K.In a similar fashion, we can select countably generated submodule T 1 and S 1 such that T 0 ⊆ S 0 ⊆ T 1 ⊆ S 1 , T 1 ∈ N and S 1 K. Continuing in this way, we obtain an ascending chain of countably generated submodules such that T n ∈ N and T n K for all n.If we take T to be the union of the chain, then T = n<ω0 T n and so T ∈ N. Also, T = n<ω0 S n implies that T K, since compatibility with K is an inductive property.Therefore, T ∩ K is a countably generated submodule from N K which contains the countable generated submodule S of K. ✷ Remark 3.1.The careful reader will observe that Theorem 3.1 remains valid, with the same proof, if M is replaced by any QT AG-module with a * -basis.It is unknown at this time whether a QT AG-module with a * -basis is necessarily totally projective.However, it is true that if M has a * -basis and H(M ) ≤ ℵ 1 , then M is totally projective (see [3]).
Remark 3.2.There exists almost totally projective modules which do not appear as isotype submodules of totally projective modules.To see this, take K to be a h-reduced almost totally projective module of length ω 0 which is not a direct sum of uniserial modules.Suppose to the contrary that K embeds as an isotype submodule in a totally projective module M .Then, , a h-reduced totally projective module of length ω 0 and hence a direct sum of uniserial modules.But this contradicts the fact that every submodule of a direct sum of uniserial modules is also a direct sum of uniserial modules.
Note that if K is a nice submodule of a QT AG-module M , then K is a separable submodule of M .This is because, for a nice submodule K, sup{H M (x+y) : y ∈ K} is actually attained by H M (x+z) for some z ∈ K. Therefore, we have the following as an immediate corollary of Theorem 3.1.
Corollary 3.1.If B is a balanced submodule of the totally projective module M , then B is almost totally projective.

The summand problem
In this section, we address the question of whether a direct summand of an almost totally projective module is almost totally projective.Even though a complete solution has so far resisted our efforts, we present two special cases in which the question can be answered affirmatively.In our first result, we consider the case when almost totally projective module appears as an isotype submodule of a totally projective module.Recall, as demonstrated in Section 3, that there exists almost totally projective modules which do not appear in this manner.Proposition 4.1.Suppose K is an isotype submodule of a totally projective module M .If K is almost totally projective, then every direct summand of K is almost totally projective.
Proof: Write K = S ⊕ T and observe that S is isotype in M .Therefore, by Theorem 3.1, it is enough to show that S is separable in M .In order to see that S is separable in M , suppose x ∈ M and observe that K is separable in M by Property (A).Thus, there is a countably generated submodule L = {y n : n < ω 0 } of K such that for every y ∈ K, there exists an n < ω 0 such that H M (x + y) ≤ H M (x + y n ).For each n, write y n = s n + t n , where s n ∈ S and t n ∈ T .We claim that if s ∈ S, there exists n < ω 0 such that H M (x + s) ≤ H M (x + s n ).Indeed, it is well known that there exists an n such that H M (x + s) ≤ H M (x + s n + t n ).So, to establish the claim and thereby complete the proof, we may assume that

✷
The following lemma is one of the main ingredient in the proof of second summand result.Lemma 4.1.Suppose M = S ⊕ T is almost totally projective.Then, M has a weak nice system N satisfying for every P ∈ N, P = (P ∩ S) ⊕ (P ∩ T ).Moreover, if H(T ) ≤ ℵ 1 , then T is totally projective.
Proof: Let N M be a weak nice system for M and let N = {P ∈ N M : P = (P ∩ S) ⊕ (P ∩ T )}.
Clearly 0 ∈ N and N is closed under ascending unions.Therefore, to establish that N is a weak nice system for M , it suffices to show that every countably generated submodule K of M is contained in a countably generated member of N.
For every x ∈ M , let x = s x + t x (s x ∈ S, t x ∈ T ) be the unique representation of x with respect to the decomposition M = S ⊕ T .Define an ascending chain of countably generated submodules from N M as follows.Take P 0 to be a countably generated submodule from N M such that P 0 ⊇ K ∪ {s x : x ∈ K} ∪ {t x : x ∈ K}.Now, if 1 ≤ n ≤ ω 0 and P n−1 has been defined, take P n to be a countably generated submodule from N M such that P n ⊇ P n−1 ∪ {s x : x ∈ P n−1 } ∪ {t x : x ∈ P n−1 }.Then P = n<ω0 P n ∈ N M contains K and is countably generated.Moreover, it is clear from the construction that P = (P ∩ S) ⊕ (P ∩ T ) so that P is actually in N.
Next suppose that H(T ) ≤ ℵ 1 and take N to be the weak nice system constructed above.Since every countably generated module is totally projective, we may assume that H(T ) = ℵ 1 .Select a smooth chain 0 = T 0 ⊆ T 1 ⊆ . . .T α ⊆ . . .(α < ω 1 ) such that β < η and α(β) < α(η).Thus, by passing to a cofinal subchain, we may assume that the P β 's and the L α(β) 's ascend.It again follows that β<µ Q β ∈ N S .✷ For a uniform element x ∈ M, e(x) = d(xR) and H M (x) = sup d yR xR | y ∈ M, x ∈ yR and y uniform are the exponent and height of x in M, respectively.H k (M ) denotes the submodule 78 A. Hasan and Rafiquddin of M generated by the elements of height at least k and H k (M ) is the submodule of M generated by the elements of exponents at most k.M is h-divisible if