# Different Characterizations of Large Submodules of QTAG-Modules.

1. Introduction and Preliminaries

All the rings R considered here are associative with unity and modules M are unital QTAG-modules. An element x [member of] M is uniform, if xR is a nonzero uniform (hence uniserial) module and, for any T-module M with a unique composition series, d(M) denotes its composition length. For a uniform element x [member of] M, e(x) = d(xR) and [H.sub.M](x) = sup {d(yR/xR) | y [member of] M, x [member of] yR and y uniform} are the exponent and height of x in M, respectively. [H.sub.k] (M) denotes the submodule of M generated by the elements of height at least k and [H.sup.k] (M) is the submodule of M generated by the elements of exponents at most k. For any arbitrary x [member of] M, H(x) = k if x [member of] [H.sub.k] (M) but x [not member of] [H.sub.k+1] (M). M is h-divisible if M = [M.sup.1] = [[intersection].sup.[infinity].sub.k=0] [H.sub.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.

A submodule N of M is h-pure in M if N [intersection] [H.sub.k] (M) = [H.sub.k] (N), for every integer k [greater than or equal to] 0. For a limit ordinal [alpha], [H.sub.[alpha]] (M) = [[intersection].sub.[rho]<[alpha]] [H.sub.[rho]] (M), for all ordinals [rho] < [alpha], and it is [alpha]-pure in M if [H.sub.[sigma]] (N) = [H.sub.[sigma]] (M) [intersection] N for all ordinals [sigma] < [alpha] and it is an isotype if it is [alpha]-pure for every ordinal [alpha]. A submodule B [subset or equal to] M is a basic submodule of M, if B is h-pure in M, B = [direct sum] [B.sub.i], where each [B.sub.i] is the direct sum of uniserial modules of length i and M/B is h-divisible. For a QTAG-module M, the [sigma]th-Ulm invariant of M, [f.sub.M] ([sigma]) is the cardinal number g(Soc([H.sub.[sigma]](M))/Soc([H.sub.[sigma]+1] (M))) [1]. Several results which hold for TAG-modules also hold good for QTAG-modules [2].

A module M is summable if Soc(M) = [[direct sum].sub.[tau]<[alpha]][S.sub.[alpha]], where [S.sub.[alpha]] is the set of all elements of [H.sub.[alpha]] (M) which are not in [H.sub.[alpha]+1] (M), where [tau] is the length of M. A QTAG-module M is called [alpha]-summable if Soc(M) = [[union].sub.n<[omega]] [M.sub.n], [M.sub.n] [subset or equal to] [M.sub.n+1] and, for every positive integer n, there is an ordinal [[alpha].sub.n] such that [mathematical expression not reproducible], [[alpha].sub.n] < length of M.

For any uniform element x [member of] M, there exist uniform elements [x.sub.1], [x.sub.2], ... such that xR [contains or equal to] [x.sub.1]R [contains or equal to] [x.sub.2]R [contains or equal to] ... and d([x.sub.i]R/[x.sub.i+1]R) = 1. Now the U7m-sequence of x is defined as U(x) = (H(x), H([x.sub.1]), H([x.sub.2]), ...). U sequences are defined as U(x). This is analogous to the U7m-sequences defined in groups [3]. These sequences are partially ordered because U(x) [less than or equal to] U(y) if H([x.sub.i]) [less than or equal to] H([y.sub.i]) for every i. For the sequence n = ([n.sub.0], [n.sub.1], [n.sub.2], ...) of nonnegative, non-decreasing integers we may consider L as the submodule of M generated by the elements x of M for which U(x) [greater than or equal to] n. If f is an endomorphism of M, then H(x) [less than or equal to] H(f(x)), and therefore L is fully invariant. Therefore with every large submodule L of M we may associate a sequence n(L).

2. Some Characterizations of Large Submodules

In this section we study and characterize the properties of fully invariant and large submodules of QTAG-modules. We also discuss the properties of large submodules inherited from the containing module.

We start with the facts which are true for any module. For a fully invariant submodule N of a QTAG-module M and an endomorphism f of M, it induces an endomorphism [bar.f] of M/N such that [bar.f] (x + N) = f(x) + N. On the other hand for the endomorphism [bar.f] of M/N induced by an endomorphism f of M and a fully invariant submodule K/N [subset or equal to] M/N, [bar.f] (x + N) = f(x) + N [member of] K/N. That is, f(x) [member of] K and K is fully invariant in M. For a fully invariant submodule A [subset or equal to] M = [direct sum] [M.sub.i], A = [direct sum] (A [intersection] [M.sub.i]) and each A [intersection] [M.sub.i] is fully invariant in [M.sub.i].

For any sequence n = ([n.sub.1], [n.sub.2], ...) we define M(n) as the submodule of M, generated by the elements x for which U(x) [greater than or equal to] n. This submodule is a large submodule of M. In fact for every large submodule there is a sequence and, for every sequence, there is a large submodule [4].

For a QTAG-module M, consider the homomorphism f: M [right arrow] M/[M.sup.1]. As [M.sup.1] = [[intersection].sup.[infinity].sub.k=0] [H.sub.k] (M), f is height preserving. This implies that H(x) = H(f(x)) and U (x) = U(f(x)) for all x [member of] M.

We conclude that L/[M.sup.1] is large in M/[M.sup.1] if and only if L is large in M. In a module M without elements of infinite height, consider a fully invariant submodule K of M, and x [member of] Soc(K) such that n = H(x) [less than or equal to] H(y) for every y [member of] Soc(K). Let z [member of] Soc(M), such that H(z) [greater than or equal to] n. Then there exists an endomorphism f of M such that f(x) = z; therefore z [member of] K and Soc(K) = Soc([H.sub.n] (M)).

Remark 1. For any large submodule L of M, Soc(L) = Soc([H.sub.n](M)) for some positive integer n.

Lemma 2. Let N be submodule of M such that [mathematical expression not reproducible], where the sequence of positive integers [n.sub.0], [n.sub.1], [n.sub.2], ... is monotonically increasing. Then M = N + B for any basic submodule B of M.

Proof. Let B = [direct sum] [B.sub.i] be a basic submodule of M and M = [B.sub.1] [direct sum] ... [direct sum] [B.sub.k] [direct sum] ([B.sup.*.sub.k], [H.sub.k] (M)) [5]. Then

[mathematical expression not reproducible]. (1)

Now suppose, for every x [member of] M, e(x) [less than or equal to] k implies that x [member of] (B + N). Consider x [member of] M such that e(x) = k + 1; then there exists y [member of] M such that d (xR/yR) = k. Now y [member of] Soc(M) and y = b + z, where b [member of] B and z [member of] Soc([H.sub.k] (N)), ensuring the existence of z' such that d(z'R/zR) = k. By the h-purity of B, there exists b' [member of] B, such that d(b'R/bR) = k. Now e(x - b' - z') [less than or equal to] k, and thus x - b' - z' [member of] B + N or x [member of] B + N implying that M = B + N.

The following remarks are significant to be stated.

Remark 3. Let L be a large submodule of an unbounded QTAG-module M without elements of infinite height M and B a proper basic submodule of M. Then

M/B = (B + L) / B [congruent to] L / (B [intersection] L); (2)

therefore L is unbounded. Conversely for an unbounded fully invariant submodule L of M, [H.sub.k] (L) is fully invariant for all k [member of] [Z.sup.+]. As an immediate consequence of Lemma 2, L is a large submodule of M. We can say that the unbounded fully invariant submodules of M are exactly the large submodules of M.

Remark 4. If [B.sub.i] is the direct sum of uniserial modules of length i and x ([not equal to] 0) [member of] [B.sub.i,] then

[mathematical expression not reproducible], (3)

where [n.sub.0] = H (x) and [n.sub.k] = [n.sub.0] + k, 0 [less than or equal to] k [less than or equal to] i - [n.sub.0] - 1. (3)

Remark 5. Let [B.sub.i] be the direct sum of uniserial modules of length i and x, y [member of] [B.sub.i]. Then there exists an endomorphism f of [B.sub.i] with f(x) = y, if and only if H(x) [less than or equal to] H(y).

Remark 6. Let A be a fully invariant submodule of [B.sub.i], a direct sum of uniserial modules of length i. Then [mathematical expression not reproducible]. If A = 0, if [n.sub.i] = i, and if A [not equal to] 0, then [n.sub.i] = min {H(x), x [member of] A}.

Remark 7. If [B.sub.i] and [B.sub.i+j] are the direct sums of uniserial modules of length i and i+j, respectively, and x [member of] [B.sub.i], y [member of] [B.sub.i+j], then

(i) there exists a homomorphism f: [B.sub.i] [right arrow] [B.sub.i+j] such that f(x) = y if and only if e(x) [greater than or equal to] e(y),

(ii) there exists a homomorphism g : [B.sub.i+j] [right arrow] [B.sub.i] such that g(y) = x if and only if H(x) [greater than or equal to] H(y).

Theorem 8. Let B = [[direct sum].sub.i][B.sub.i], where each [B.sub.i] is the direct sum of uniserial modules of length i. Then L is a fully invariant submodule of B if and only if [mathematical expression not reproducible]. A fully invariant submodule L is large in B if and only if [mathematical expression not reproducible]; the above conditions hold and the sequence <1 - [n.sub.1], 2 - [n.sub.2], 3 - [n.sub.3], ...> is unbounded if B is unbounded.

Proof. Let L be a fully invariant submodule of B. Then

[mathematical expression not reproducible] (4)

by the facts mentioned above and Remark 6. Now [n.sub.i] [less than or equal to] i for i [member of] [Z.sup.+] and the first condition holds. If L = 0, then [mathematical expression not reproducible] for every i; therefore [n.sub.i] = i for every i and the second condition holds. If L [not equal to] 0, then there exists a least positive integer k such that [mathematical expression not reproducible]. Then [mathematical expression not reproducible] for all i [greater than or equal to] k, where [B.sub.i] [not equal to] 0. Since Soc([B.sub.k]) = Soc([H.sub.k-1] ([B.sub.k])) [subset or equal to] L, this implies that Soc([H.sub.k-1] (B)) [subset or equal to] Soc(L). Again [mathematical expression not reproducible]. Now suppose L [not equal to] 0 and [B.sub.i] [not equal to] 0 = [B.sub.i+j]. If [mathematical expression not reproducible], then [mathematical expression not reproducible] and the second condition holds. We assume that [mathematical expression not reproducible]. Consider x [member of] [B.sub.i] such that H(x) [greater than or equal to] [n.sub.i+j] and [mathematical expression not reproducible] such that H(y) = [n.sub.i+j]. Now, by Remark 7, there exists an endomorphism g of B mapping y onto x. Hence x [member of] L and [mathematical expression not reproducible]; thus [n.sub.i] [less than or equal to] [n.sub.i+j].

Now suppose [mathematical expression not reproducible]. Then [n.sub.i] = i so [n.sub.i+j] [less than or equal to] i + j = [n.sub.i] + j. If [mathematical expression not reproducible] such that H(y) [greater than or equal to] [n.sub.i] + j, we may choose x [member of] [B.sub.i] such that H(x) = [n.sub.i]. Then e(x) = i - [n.sub.i] and e(y) [less than or equal to] i + j - ([n.sub.i] + j) = i - [n.sub.i]. By Remark 7, there exists an endomorphism f of B with f(x) = y. Thus y [member of] L and we have [mathematical expression not reproducible]; therefore [n.sub.i+j] [less than or equal to] [n.sub.i] + j.

If [B.sub.i] [not equal to] 0 [not equal to] [B.sub.i+j], then [n.sub.i] [less than or equal to] [n.sub.i+j] [less than or equal to] [n.sub.i] + j but if [B.sub.i] = 0, we may define [n.sub.i] so that this inequality holds for all i. Thus all fully invariant submodules of B are the direct sums of [mathematical expression not reproducible]. If L is a large submodule of B and B is unbounded, then, by Lemma 2, L is also unbounded. Therefore <1 - [n.sub.1], 2 - [n.sub.2], 3 - [n.sub.3], ...> must be unbounded.

For the converse, suppose [mathematical expression not reproducible]. To establish the full invariance of L, we consider any i [member of] [Z.sup.+] and [mathematical expression not reproducible]. We have to show that for any endomorphism f of B, f(x) [member of] L. Consider x [not equal to] 0, such that f(x) = [x.sub.1] + ... + [x.sub.l], where [x.sub.r] [member of] [B.sub.r] and H(x) [less than or equal to] H(f(x)) = min (H([x.sub.k])), 1 [less than or equal to] k [less than or equal to] l, e(x) [greater than or equal to] e(f(x)) = max {e([x.sub.k]) | 1 [less than or equal to] k [less than or equal to] l}. If [mathematical expression not reproducible]. Thus [mathematical expression not reproducible].

This implies that L is a fully invariant submodule of B. If B is unbounded and <1 - [n.sub.1], 2 - [n.sub.2], 3 - [n.sub.3], ...) is also unbounded, then L is unbounded and is therefore a large submodule of B by Remark 3.

Corollary 9. If L is a large submodule of a QTAG-module M, then M/L is a direct sum of uniserial modules.

Proof. For any basic submodule B of M,

[mathematical expression not reproducible] (5)

and the result follows.

Corollary 10. For any large submodule L of M, [L.sup.1] = [M.sup.1].

Proof. Since M/L is a direct sum of uniserial modules, [(M/L).sup.1] = 0 or [M.sup.1] = [L.sup.1].

Theorem 11. Let N be h-pure submodule of a QTAG-module M and L a large submodule of N. Then there exists a large submodule L' of M such that L' [intersection] N = L. If M/N is h-divisible, then L' is the closure of L in M and is therefore uniquely determined by L and M/L' [congruent to] N/L.

Proof. Let L = N(n) and L' = M(n). Since N is h-pure in M and n = <[n.sub.1], [n.sub.2], ...> is a U-sequence for N, we have that n is a U-sequence for M. Thus L' is a large submodule of M.

If x [member of] L, then [U.sub.M] (x) = [U.sub.N] (x) [greater than or equal to] n; therefore x [member of] L' [intersection] N and L [subset or equal to] L' [intersection] N. Conversely if y [member of] L' [intersection] N, then [U.sub.N] (y) = [U.sub.M] (y) [greater than or equal to] n implies that y [member of] L or L' [intersection] N [subset or equal to] L. Thus L = L' [intersection] N.

Let M/N be h-divisible and L' a large submodule of M with L' [intersection] N = L. Then

L'/L = L'/(L' [intersection] N) [congruent to] (N + L') / N = M/N. (6)

That is, L'/L is h-divisible. But M/L' [congruent to] (M/L)/(L'/L), where L'/L is a direct summand of M/L; we have M/L [congruent to] (L'/L) [direct sum] (M/L') and M/L' is a direct sum of uniserial modules [6]. Now M/L' [congruent to] N/L, thus

M/L' = (N + L') / L' [congruent to] N / (N [intersection] L') = N/L. (7)

Now we characterize large submodules in terms of Ulm invariants.

Theorem 12. Let L be a submodule of a QTAG-module M. Then L is a large submodule of M if and only if [mathematical expression not reproducible], where

(i) [n.sub.k] [less than or equal to] k, k [member of] [Z.sup.+],

(ii) [n.sub.k] [less than or equal to] [n.sub.k+1] [less than or equal to] [n.sub.k] + 1,

(iii) the sequence <1 - [n.sub.1], 2 - [n.sub.2], 3 - [n.sup.3], ...> is unbounded if M is unbounded and the Ulm-invariants of L are given by [f.sub.L] (n) = [[summation].sub.k] ([f.sub.M] (k - 1)), k - [n.sub.k] -1 = n, for all n [member of] [Z.sup.+].

Proof. Suppose [mathematical expression not reproducible]. Since [mathematical expression not reproducible] are fully invariant submodules, their sum is again fully invariant submodule of M. If M is bounded, then L is large. If M is unbounded, then, by the third condition, for each j [member of] [Z.sup.+], there exists a positive integer i such that i - [n.sub.i] > j or i > [n.sub.i] + j.

Since, [mathematical expression not reproducible], If [mathematical expression not reproducible]. Now [mathematical expression not reproducible], where i - [n.sub.i] > j or i - [n.sub.i] [greater than or equal to] j + 1; thus i [greater than or equal to] [n.sub.i] + j + 1 and y [member of] [H.sup.i] (M).

If d(yR/zR) = [n.sub.i], then [mathematical expression not reproducible] because d(yR/xR) = j + [n.sub.i] and d(zR/xR) = j. Now, by Lemma 2, L + B = M, for every basic submodule B of M, and L is a large submodule of M.

Conversely suppose L is a large submodule of M. Then for any basic submodule B of M, L [intersection] B is a large submodule of B and, by Theorem 8, [mathematical expression not reproducible] satisfy the given conditions.

Now,

[mathematical expression not reproducible] (8)

and, for each j [member of] [Z.sup.+],

[mathematical expression not reproducible]. (9)

This implies that [mathematical expression not reproducible].

For the converse, consider [mathematical expression not reproducible]. Then [H.sub.i] ([x.sub.i]) [greater than or equal to] H(x) [greater than or equal to] [n.sub.j] for [x.sub.i], 1 [less than or equal to] i [less than or equal to] m and e([x.sub.i]) [less than or equal to] e(x) [less than or equal to] j - [n.sub.i] for all [x.sub.i]'s. Now, for [mathematical expression not reproducible]. If i = j + l for l [member of] [Z.sup.+], then e([x.sub.i]) < j - [n.sub.j] - j + l - ([n.sub.j] + 1) [less than or equal to]

j + l - [n.sub.j+l]. (by the given condition). Therefore

[mathematical expression not reproducible] (10)

and [mathematical expression not reproducible]. Let [mathematical expression not reproducible]. Now [mathematical expression not reproducible]. Since B is h-pure in M and M/B is h-divisible, L' = L, by Theorem 11. Again L [intersection] B is a basic submodule of L; thus [f.sub.L] (n) = [f.sub.L[intersection]B] (n), for all n [member of] [Z.sup.+].

If L [intersection] B = [direct sum] [(L [intersection] [B.sub.i]).sub.i], where [(L [intersection] B).sub.i] is the direct sum of uniserial modules of length i, then [mathematical expression not reproducible], is a direct sum of uniserial modules of length n+1.

Again,

[mathematical expression not reproducible]. (11)

And the proof is complete.

3. Properties of Large Submodules of QTAG-Modules

In this section we compare the structures of QTAG-modules and their large submodules. We investigate the characteristics of QTAG-modules which are preserved by their large submodules. We start with the [SIGMA]-modules, that is, the modules whose high submodules are direct sums of uniserial modules [7]. Then we study summable, [sigma]-summable, ([omega] + 1)-projective, and h-pure complete QTAG-modules.

Singh [8] proved that a QTAG-module M is a direct sum of uniserial submodules if and only if M is the union of an

ascending sequence of submodules [M.sub.n], n = 1, 2, 3, ..., such that, for every n, there exists [k.sub.n] > 0 and [H.sub.M] (x) [less than or equal to] [k.sub.n] for all x [member of] [M.sub.n].

This helps us to prove the following.

Theorem 13. A QTAG-module is a [SIGMA]-module if and only if Soc(M) = [[union].sup.[infinity].sub.k=1] [M.sub.k], where [M.sub.k] [subset] [M.sub.k+1] and for every k [member of] N, [M.sub.k] [intersection] [H.sub.k] (M) = Soc([M.sup.1]).

Proof. Since M is a [SIGMA]-module, it contains a high submodule N such that N is a direct sum of uniserial modules.

Again N is a high submodule [9] of M if and only if N is h-pure in M and Soc(M) = Soc(N) + Soc([M.sup.1]). Therefore by the above result [8], Soc(N) = [[union].sup.[infinity].sub.k=1] [N.sub.k], [N.sub.k] [subset or equal to] [N.sub.k+1], and [N.sub.k] [intersection] [H.sub.k] (N) = 0, and we deduce Soc(M) = [[union].sup.[infinity].sub.k=1] ([N.sub.k] + Soc([M.sup.1])). If we put [M.sub.k] = Soc([M.sup.1]) + [N.sub.k], then [M.sub.k] [subset or equal to] [M.sub.k+1] and (Soc([M.sup.1]) + [N.sub.k]) [intersection] Soc([M.sup.1]) + Soc([H.sub.k] (N)) - Soc([M.sup.1]) + ([N.sub.k] [intersection] [H.sub.k] (N)) = Soc([M.sup.1]), because [H.sub.k] (N) is a high submodule of [H.sub.k] (M).

For the converse if Soc(N) = [[union].sup.[infinity].sub.k=1] ([M.sub.k] [intersection] N) = [[union].sup.[infinity].sub.k=1] [N.sub.k], where we put [N.sub.k] = N [intersection] [M.sub.k], then [N.sub.k] [subset or equal to] [N.sub.k+1]. Also

[mathematical expression not reproducible]. (12)

Therefore N is a direct sum of uniserial modules and M is a [SIGMA]-module.

Now we may prove the following.

Theorem 14. A QTAG-module M is a [SIGMA]-module if and only if its large submodule L is a [SIGMA]-module.

Proof. Since [L.sup.1] = [M.sup.1] [6], there is a natural number m such that [mathematical expression not reproducible]. If M is a [SIGMA]-module, then, by Theorem 13, Soc(M) is the union of ascending chain of submodules [M.sub.k] such that [M.sub.k] [subset or equal to] [M.sub.k+1] and [M.sub.k] [intersection] [H.sub.k] (M) = Soc([M.sup.1]) for every k [member of] N.

This implies that Soc(L) = [U.sub.k<[omega]] ([M.sub.k] [intersection] L) and [M.sub.k] [intersection] L [subset or equal to] [M.sub.k+1] [intersection] L. Therefore

[mathematical expression not reproducible]. (13)

Now Theorem 13 indicates that L is a [SIGMA]-module. Conversely suppose L is a [SIGMA]-module. Therefore

[mathematical expression not reproducible]. (14)

Again Soc([H.sub.m] (M)) = [[union].sub.n<[omega]] [L.sub.n]. Now

[mathematical expression not reproducible]. (15)

Thus, by Theorem 13, [H.sub.m] (M) is a [SIGMA]-module, and so is M.

To study the other relations between a module M and its large submodule L we need the following lemma.

Lemma 15. Isotype submodules of countable length of summable QTAG-modules are again summable.

Proof. Let N be an isotype submodule of countable length [rho] in the summable module M. Now there is a [H.sub.[rho]] (M)-high submodule K of M such that N [subset or equal to] K. Since Soc (M) = [[direct sum].sub.[sigma]<[rho]] [S.sub.[sigma]], there is [H.sub.[rho]] (M)-high submodule P of M such that Soc(P) = [[direct sum].sub.[sigma]<[rho]] [S.sub.[sigma]].

Again, for every ordinal [rho], every [H.sub.[rho]] (M)-high submodule is isotype; therefore P is isotype and it is summable. The socles of [H.sub.[rho]] (M)-high submodules have the same images under the canonical map M [right arrow] M/[H.sub.[rho]] (M) because this maps [H.sub.[rho]] (M)-high submodules isomorphically in a height preserving manner onto submodules of M/[H.sub.[rho]] (M).

Now N is isotype in a summable module K of countable length [rho]. Therefore Soc(K) is the union of an ascending chain of submodules [K.sub.n], where for every n the heights of elements of [K.sub.n] assume but a finite numbers of values.

Now Soc(N) = [union] [K.sub.n] [intersection] N, n = 1, 2, 3, ..., and the heights of the elements of [K.sub.n] [intersection] N assume a finite numbers of different values. Thus N is summable.

The following result shows that summability is shared by large submodules.

Theorem 16. Let L be a large submodule of a QTAG-module M. Then M is summable if and only if L is summable.

Proof. Suppose M is summable; that is, Soc(M) = [[direct sum].sub.[beta]<[alpha]] [M.sub.[beta]], where the nonzero elements of [M.sub.[beta]]'s are contained in [H.sub.[beta]] (M) but they do not belong to [H.sub.[beta]+1] (M), for every [beta] < [alpha].

Again L is fully invariant submodule of M and [H.sub.[rho]] (M) = [H.sub.[rho]] (L) for all ordinals [rho] [greater than or equal to] [omega], Soc(L) = [[direct sum].sub.[beta]<[alpha]] ([M.sub.[beta]] [intersection] L), where the nonzero elements of [M.sub.[beta]] [intersection] L are contained in [H.sub.[beta]] (L) and not contained in [H.sub.[beta]+1] (L) for every [omega] [less than or equal to] [beta] < [alpha]. Since [mathematical expression not reproducible], whenever 1 [less than or equal to] n < [omega], n [less than or equal to] [t.sub.n] < [omega], [M.sub.[beta]] [intersection] L [subset or equal to] L, but ([M.sub.[beta]] [intersection] L) [subset or equal to] [H.sub.1] (L) = 0, for each [beta] < [t.sub.1]. By transfinite induction [M.sub.[beta]] [intersection] L [subset or equal to] [H.sub.1] (L) and ([M.sub.[beta]] [intersection] L) [intersection] [H.sub.2] (L) = 0, for [t.sub.1] [less than or equal to] [beta] < [t.sub.2] and so on; that is, [H.sub.[beta]] [intersection] L [subset or equal to] [H.sub.n] (L) and ([H.sub.[beta]] [intersection] L) [intersection] [H.sub.n+1] (L) = 0, for [t.sub.n] [less than or equal to] [beta] < [t.sub.n+1].

If we put [mathematical expression not reproducible]. Therefore L is summable.

Conversely suppose L is summable. So, [L.sup.1] = [M.sup.1] is summable as its fully invariant submodule. Moreover, by Lemma 15, L being summable implies that L is a [SIGMA]-module. Now by Theorem 14, M is also a [SIGMA]-module. For a high submodule N of M, Soc(N) [direct sum] Soc([M.sup.1]) = Soc(M).

Since N is a direct sum of uniserial modules, Soc(N) = [[direct sum].sub.k<[omega]] [N.sub.k], where [N.sub.k] [subset or equal to] [H.sub.k] (M) and [N.sub.k] [intersection] [H.sub.k+1] (M) = 0 because N is h-pure in M. Again the summability of [M.sup.1] ensures that Soc([M.sup.1]) = [[direct sum].sub.[beta]<[alpha]][K.sub.[beta]], where [K.sub.[beta]] [subset or equal to] [H.sub.[beta]] ([M.sup.1]) and [K.sub.[beta]] [intersection] [H.sub.[beta]+1] ([M.sup.1]) = 0. Therefore, [K.sub.[beta]] [subset or equal to] [H.sub.[omega]+[beta]] (M) and [K.sub.[beta]] [intersection] [H.sub.[omega]+[beta]+1] (M) = 0. This implies that

[mathematical expression not reproducible]. (16)

We may infer now that M is summable.

Theorem 17. Let L be the large submodule of M. Then M is [sigma]-summable if and only if L is [alpha]-summable.

Proof. Suppose M is unbounded. Then length of M = length of L [greater than or equal to] [omega]. If M is [sigma]-summable, then L is also [sigma]-summable being a submodule of equal length.

If M is bounded the result holds trivially.

Conversely suppose L is [sigma]-summable. Therefore [mathematical expression not reproducible] for all n [greater than or equal to] 0 and some [[alpha].sub.n] < length of M.

Now, [mathematical expression not reproducible] both. Thus [mathematical expression not reproducible], whenever [s.sub.n] < length of M = length of [H.sub.m] (M) [greater than or equal to] [omega], [s.sub.n] = [[alpha].sub.n] [greater than or equal to] [omega] or [omega] > [s.sub.n] = [k.sub.n].

We may define [M.sub.n] = {x | x [member of] Soc(M) [intersection] [L.sub.n] and x [not member of] [H.sub.m] (M)}. Thus Soc (M) = [[union].sub.n<[omega]] [M.sub.n], [M.sub.n] [subset or equal to] [M.sub.n+1]. By defining [M.sub.n]'s we observe that [mathematical expression not reproducible]. This implies that M is [sigma]-summable.

Theorem 18. If M is a direct sum of [sigma]-summable QTAG-modules, then so is L.

Proof. Let M = [[direct sum].sub.i[member of]I] [M.sub.i], where each [M.sub.i] is [sigma]-summable. Now L = [[direct sum].sub.i[member of]I] (L [intersection] [M.sub.i]) because L is fully invariant in M. Since all [M.sub.i]'s are isotype in M, we infer that L [intersection] [M.sub.i] is large in [M.sub.i], for every i. By Theorem 17, L [intersection] [M.sub.i] are [sigma]-summable. Thus L is also a direct sum of [sigma]-summable modules.

Let us recall the following.

Definition 19. A QTAG-module M is ([omega] + 1)-projective if there exists a submodule N [subset or equal to] Soc(M) such that M/N is a direct sum of uniserial modules.

Remark 20. The submodules of ([omega] + n)-projective modules are also ([omega] + n)-projective.

Theorem 21. A QTAG-module M is ([omega] + 1)-projective if and only if its large submodule L is ([omega] + 1)-projective.

Proof. Suppose L is ([omega] + 1)-projective. Therefore there exists a submodule N [subset or equal to] Soc(L) such that Soc(L/N) = [[union].sub.n<[omega]] ([L.sub.k]/N), where [L.sub.k] [subset or equal to] [L.sub.k+1] [subset or equal to] L and [L.sub.k] [intersection] [H.sub.k] (L) [subset or equal to] N for each k < [omega]. Now [L.sub.k] [subset or equal to] [H.sup.2] (M), for every k < [omega]. Since

[mathematical expression not reproducible], (17)

for some k [less than or equal to] [j.sub.k] [less than or equal to] [t.sub.k] < [omega], we have

[mathematical expression not reproducible]. (18)

Therefore the heights of the elements of [L.sub.k]/N are bounded in M/N for all k < [omega]. Now (M/N)/(L/N) [congruent to] M/L is a direct sum of uniserial modules [6]. Therefore M/N is a direct sum of uniserial modules and M is ([omega]+1)-projective. The converse is trivial.

The property of being h-pure complete is also shared by the large submodules of QTAG-modules.

First we recall the definition of h-pure completeness.

Definition 22. A QTAG-module M is h-pure complete if, for every subsocle S [subset or equal to] Soc(M), there is a h-pure submodule N of M so that S = Soc(N). In other words every subsocle supports a h-pure submodule of M.

Theorem 23. Let L be the large submodule ofa QTAG-module M. If M is h-pure complete, so is L.

Proof. Let S be a subsocle of L. Since S [subset or equal to] Soc(M), S supports a h-pure submodule N of M. Now N [intersection] L is also large in M and N [intersection] L is h-pure in L. Again S = Soc(N) [intersection] Soc(L) = Soc(N [intersection] L), and therefore L is h-pure complete.

Corollary 24. A QTAG-module M is h-pure complete if and only if [H.sub.k] (M) is h-pure complete for some fixed but arbitrary positive integer k.

Proof. Since [H.sub.k] (M) is large in M, it is h-pure complete if M is h-pure complete. Conversely suppose [H.sub.k] (M) is h-pure complete. We shall use transfinite induction to prove the result.

Let S be a subsocle of M such that S [intersection] [H.sub.1] (M) [subset or equal to] Soc([H.sub.1] (M)) and S [intersection] [H.sub.1] (M) = Soc(N) for some h-pure submodule N of [H.sub.1] (M). By [7] we can say that there is a h-pure submodule K of M such that [H.sub.1] (K) = N and Soc(K) = Soc(N) = Soc([H.sub.1] (K)). Now S [intersection] [H.sub.1] (M) = Soc([H.sub.1] (K)).

We have to show that there exists a h-pure submodule T [subset or equal to] M such that S = Soc(T). We define the submodule T = K + (S [intersection] [H.sub.2] (M)). Now

Soc(T) = Soc(K + S [intersection] [H.sub.2] (M)) = Soc (K) + (S [intersection] [H.sub.2] (M)) (19)

because S [intersection] [H.sub.2] (M) = Soc(S [intersection] [H.sub.2] (M)). Again Soc(T) = Soc(N) + (S [intersection] [H.sub.2] (M)) [subset or equal to] S.

Now

[mathematical expression not reproducible]. (20)

Therefore S = Soc(T). Now,

[mathematical expression not reproducible]. (21)

This implies that T is h-pure in M.

In the end we state the following unsolved problems.

Problem 25. Is it true that M is a HF-module if and only if its large submodule L is?

Problem 26. Is it true that M is a direct sum of closed modules if and only if its large submodule L is?

http://dx.doi.org/ 10.1155/2017/2496246

Competing Interests

The authors declare that they have no competing interests.

References

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[4] A. Mehdi, S. A. R. K. Naji, and A. Hasan, "Small homomorphisms and large submodules of QTAG- modules," Scientia Series A: Mathematical Sciences, vol. 23, pp. 19-24, 2012.

[5] M. Z. Khan, "On basic submodules," Tamkang Journal of Mathematics, vol. 10, no. 1, pp. 24-29, 1979.

[6] A. H. Ansari, M. Ahmad, and M. Z. Khan, "Some decomposition theorems on S2-module III," Tamkang Journal of Mathematics, vol. 12, no. 2, pp. 147-154, 1981.

[7] M. Z. Khan, "Modules behaving like torsion abelian groups. II, " Mathematica Japonica, vol. 23, no. 5, pp. 509-516, 1979.

[8] S. Singh, "Some decomposition theorems on abelian groups and their generalisations. II," Osaka Journal of Mathematics, vol. 16, no. 1, pp. 45-55, 1979.

[9] A. Mehdi and F. Mehdi, "N-high submodules and h-topology, " The South East Asian Journal of Mathematics and Mathematical Sciences, vol. 1, no. 1, pp. 83-88, 2002.

Fahad Sikander, (1) Alveera Mehdi, (2) and Sabah A. R. K. Naji (3)

(1) College of Science and Theoretical Studies, Saudi Electronic University, Jeddah Branch, Jeddah 23442, Saudi Arabia

(2) Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

(3) Department of Mathematics, Al Bayda University, Al Bayda, Yemen

Correspondence should be addressed to Alveera Mehdi; amehdi.wc@amu.ac.in

Received 24 July 2016; Accepted 5 December 2016; Published 3 January 2017