1 Introduction and preliminaries

The concepts for groups such as projectivity, purity, height etc carried to modules preserving the sense. For getting those results of groups, which does not hold for modules; some constraints applied on the structure of module or underlying ring. After taking into effects these constraints for the QTAG-module structure, numerous natural facts of groups can be established that are not true generally. The findings in [10] serve as the inspiration for the outcomes in this work.

The study on the structure of QTAG-modules was started by Singh [11]. After that many researchers such as Khan, Mehdi, Abbasi etc. generalized different concepts of groups to QTAG-modules [4, 9] etc. They introduced various notions and structures for QTAG-modules motivated from group structure and obtained some exciting results. Yet many concepts remain to generalize for modules.

Some of the fundamental definitions used in this manuscript have already appeared in one of the co-authors’ previous works; these are offered as quotations and are duly cited here.

“A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules [12]. All the rings R considered here are associative with unity and modules M are unital QTAG-modules. An element \(x\in M\) is uniform, if xR is a non-zero uniform (hence uniserial) module and for any R-module M with a unique composition series, d(M) denotes its composition length. For a uniform element \(x\in M,~e(x)=d(xR)\) and \(H_M(x)=\text{ sup }\left\{ d\left( \displaystyle {\frac{yR}{xR}}\right) \mid y\in M,~x\in yR~ \text{ and }~ y~ \text{ uniform }\right\} \) are the exponent and height of x in M,  respectively. \(H_k(M)\) denotes the submodule 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 \(M=M^1=\bigcap \nolimits _{k=0}^\infty ~H_k(M)\) [9] 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 QTAG-module M is said to be separable, if \(M^1=0.\) If \(M, M'\) are QTAG-modules then a homomorphism \(f:M\rightarrow M'\) is an isometry if it is 1-1, onto and \(H_{M'}(f(x))=H_M(x)\), for all \(x\in M.\) A submodule N of a QTAG-module M is a nice submodule if every nonzero coset \(a+N\) is proper with respect to N i.e. for every nonzero \(a+N\) there is an element \(b\in N\) such that \(H_M(a+b)=H_{M/N}(a+N).\)”

“A family \({\mathcal {N}}\) of submodules of M is called a nice system in M if

  1. (i)

    \(0\in {\mathcal {N}}\);

  2. (ii)

    If \(\{N_i\} _{i\in I}\) is any subset of \({\mathcal {N}},\) then \(\Sigma _I N_i \in {\mathcal {N}};\)

  3. (iii)

    Given any \(N \in {\mathcal {N}}\) and any countable subset X of M,  there exists \(K \in {\mathcal {N}}\) containing \(N\cup X\), such that K/N is countably generated [4].

Every submodule in a nice system is nice submodule. A h-reduced QTAG-module M is called totally projective if it has a nice system and direct sums and direct summands of totally projective modules are also totally projective. A submodule N of M is h-pure in M if \(N\cap H_k(M)=H_k(N),\) for every integer \(k\ge 0.\) A QTAG- module M is \((\omega + n)\)- projective, if there exists a submodule \(N \subset H^n(M)\) such that M/N is a direct sum of uniserial modules or equivalently, if and only if there is a direct sum of uniserial module K with a submodule \(L \subseteq H^n(K)\) such that \(M\cong K/L\). M is \(\omega \)-projective if and only if it is a direct sum of uniserial modules. Also two \((\omega + n)\)-projective QTAG-modules \(M_1~ , M_2\) are isometric if and only if there is a height preserving isomorphism between \(H^n(M_1)\) and \(H^n(M_2)\) [4]. For any QTAG- module M,  g(M) denotes the smallest cardinal number \(\lambda \) such that M admits a generating set X of uniform elements of cardinality \(\lambda ~ i.e.\), \(|X|=\lambda \). A homomorphism \(f : M \rightarrow N\) is said to be \(\omega _1\)-bijective if \(g(ker~f),~ g(N/f(M))<\omega _1.\)”

2 Main results

Mehdi et al. defined quasi isomorphic modules [5], studied their properties and quasi-isomorphic invariants. First we recall the definitions of \((\omega +n)\)-projective, HT-modules, \(\omega _1\)-separable modules.

“A QTAG-module M is \((\omega +n)\)-projective if and only if there exists a submodule \(N\subset H^n(M)\) such that M/N is a direct sum of uniserial modules. M is a HT-module [6] if every homomorphism from M to N is small, where N is a direct sum of uniserial modules. Equivalently M is a HT-module if and only if \(N\supset Soc(H_k(M))\) for some \(k<\omega \) whenever M/N is a direct sum of uniserial modules. A separable QTAG-module M is weakly \(\omega _1\)-separable [1] if and only if for every countably generated submodule \(N\subset M\), its closure \( {\overline{N}}\) is also countably generated i, e. \({{\overline{N}}}=\bigcap \nolimits _{k<\omega }{(N+H_k(M))}\). Two QTAG modules

are isomorphic if there exists submodules \(N\subset M\), \(N'\subset M'\) and integers n, k such that \(N\subset H_n(M)\), \(N'\subset H_k(M')\) and \(N\cong N'\).”

Here we investigate the modules M and \(M'\) when \(H_k(M)\) is isomorphic to a submodule of \(M'\). Equivalently we consider the submodule \(N\subset M\) and \(H_k(M)\subseteq N\).

Now we are able to prove the following:

Theorem 1

Let M be a QTAG-module with a submodule N such that \(H_n(M)\subseteq N\), for some \(n<\omega \). Then M is summable if and only if N is summable.

Proof

Suppose M is summable. Now \(H_n(M)\) and \(H_\omega (M)\) both are summable and \(H_\omega (M)=H_\omega (N)\). Since M and \(H_n(M)\) are summable they are \(\Sigma \)-modules as they are fully invariant submodules [8]. Therefore N is also a \(\Sigma \) -module. Let K be a high submodule of N. Therefore \(Soc(N)=Soc(K)\oplus Soc(H_\omega (N))\). Since K is a \(\Sigma \) -module we may write \(Soc(K)=\bigoplus \nolimits _{i<\omega }Soc (K_i), K_i\backslash \{0\}\subseteq H_i(N)\backslash H_{i+1}(N)\) because \(K_i\)’s are h -pure in N. Since \(H_\omega (N)\) is summable \(Soc(H_\omega (N))=\bigoplus \nolimits _{\rho <\sigma }N_\rho \), where for each \(\rho <\sigma \), \(N_\rho \backslash \{0\}\subseteq H_{\omega +\rho }(N)\backslash H_{\omega +\rho +1}(N)\). Therefore \(Soc(N)=\bigoplus \nolimits _{i<\omega }K_i\oplus (\bigoplus \nolimits _{\rho<\sigma }(N_\rho )=\bigoplus \nolimits _{\mu <\omega +\sigma }T_\mu \), where \(T_i=K_i\) if \( i<\omega \) and \(T_{\omega +\rho }=N_\rho \) whenever \(0\le \rho <\sigma \). Now \(T_\mu \backslash \{0\}\subseteq H_\mu (N)\backslash H_{\mu +1}(N)\) for all \(\mu <\omega +\sigma \) implying that N is summable.

For the converse suppose N is summable. Now N is a \(\Sigma \)-module and by [5, 7]; M is a \(\Sigma \) -module. Since N is summable \(H_\omega (N)=H_\omega (M)\) is summable, therefore M is also summable. \(\square \)

Theorem 2

Let M be a QTAG-module with a submodule N such that \(H_n(M)\subseteq N\) for some \(n<\omega \). Then N is \(\sigma \) -summable if and only if M is \(\sigma \) -summable.

Proof

If M is \(\sigma \) -summable, then by [1, 8]; \(Soc(M) = \bigcup \nolimits _{k<\omega }M_k\), \(M_k\subseteq M_{k+1}\) and for all \(k<\omega \), there exists an ordinal \(\alpha _k<\alpha \) such that \(M_k\cap H_{\alpha _k}(M)=0\), where \( \alpha \) is the length of M.Therefore \(Soc(N)=\bigcup \nolimits _{k<\omega }{(M_k\cap N)}\) and \(M_k\cap N\subseteq M_{k+1}\cap N\). Since \(H_\omega (N)=H_\omega (M)\), length of (N)=length of (M), if length of \(M \ge \omega \). Also \(M_k\cap N\cap H_{\alpha _k}(N)\subseteq M_k\cap H_{\alpha _k}(M)=0\) whenever \(\alpha _k< length (N)\) . If M is bounded then so is N and we are done.

Conversely suppose N is \(\sigma \)-summable i.e. \(Soc(N)=\bigcup \nolimits _{j<\omega }N_j\) ,\(N_j\subset N_{j+1}\) and there exist ordinals \(\alpha _j\) such that \(N_j\cap H_{\alpha _j}(N)=0\) for all \(j\ge 0\) and \(\alpha _j< length (N)=length(M)\ge \omega \). If length of \(M<\omega \) then we are done, otherwise \(Soc( H_n(M))=\bigcup (N_j\cap H_n(M))\). Now \(N_j\cap H_n(M)\subseteq N_{j+1}\cap H_n(M)\) and \(N_j\cap H_n(M)\cap H_{n+{\alpha _j}}(M)=N_j\cap H_{n+\alpha _j}(M)\subseteq N_j\cap H_{\alpha _j}(N)=0\) where \(n+\alpha _j<length (M)\). Since the length of M is cofinal with \(\omega \), \(H_n(M)\) is \(\sigma \) -summable. Therefore M is also \(\sigma \) -summable. \(\square \)

Theorem 3

Let M be a QTAG module with a submodule N such that \(H_k(M)\subseteq N\) for some \(k<\omega \). Then M is \((\omega +n)\) -projective if and only if N is \((\omega +n)\) -projective for some \(n<\omega \).

Proof

Suppose N is \((\omega +n)\)-projective. Now there exists a submodule \(K\subseteq H^n(N)\) such that N/K is a direct sum of uniserial modules. Therefore \(\displaystyle \frac{ H_k(M)+K}{K}\subseteq \displaystyle \frac{N}{K}\) is also a direct sum of uniserial modules. Since \(H_k(M/K)=\displaystyle \frac{ H_k(M)+K}{K}\), \(\displaystyle \frac{M}{K}\) is a direct sum of uniserial modules. Since \(K\subseteq H^n(M)\), M is \((\omega +n)\) -projective. Conversely if M is \((\omega +n)\) -projective then being a submodule of M, N is also \((\omega +n)\)-projective [3]. \(\square \)

Theorem 4

Let M be a QTAG module with a submodule N such that \(H_k(M)\subseteq N\) for some \(k<\omega \). Then M is a strong \(\omega \) -elongation of a totally projective module by a \((\omega +n)\) -projective module if and only if N has the same property.

Proof

Let M be a strong \(\omega \)-elongation of a totally projective module by a \((\omega +n)\)-projective module. Now \(H_{\omega }(M)\) is totally projective, therefore there exists a submodule \(K\subseteq H^n(M)\) such that \(\displaystyle \frac{M}{K+H_\omega (M)}\) is a direct sum of uniserial modules. Since \(H_k(M)\subset N\) and \(H_\omega (M)=H_\omega (N)\), \(H_\omega (N)\) is totally projective. Now \(\displaystyle {\frac{M}{K+H_\omega (M)}}\supseteq \displaystyle {\frac{N+K}{K+H_\omega (M)}}=\displaystyle {\frac{N+K+H_\omega (N)}{K+H_\omega (N)}}\simeq \displaystyle \frac{N}{N\cap (K+H_\omega (N))} =\displaystyle \frac{N}{H_\omega (N)+(N\cap K)}\) is a direct sum of uniserial modules. Since \(N\cap K\subseteq H^k(N)\), we are done.

For the converse suppose N is a strong \(\omega \)-elongation of a totally projective module by a \((\omega +n)\)-projective module. Now \(H_\omega (N)=H_\omega (M)\) is totally projective. Moreover, if \(K\subseteq H^k(N)\) with \(\displaystyle {\frac{N}{K+H_\omega (N)}}\), a direct sum of uniserial modules, we have \(\displaystyle \frac{M/(K+H_\omega (N))}{N/(K+H_\omega (N))}\) is bounded. Therefore, \(\displaystyle {\frac{M}{K+H_\omega (N)}}=\displaystyle {\frac{M}{K+H_\omega (M)}}\) is a direct sum of uniserial modules as \(K\subseteq H^k(M)\) and the result follows.\(\square \)

For the QTAG modules M, \(M'\) a homomorphism \( f:M \longrightarrow M'\) is said to be small if ker f contains a large submodule. Naji [6] defined HT-modules with respect to a family \({\mathcal {F}}\) of QTAG-modules. A QTAG-module is a HT-module with respect to a family \({\mathcal {F}}\) of QTAG-modules if \(Hom(M,K)=Hom_s(M,K)\) for every \(K\in {\mathcal {F}}\), whenever K is a direct sum of uniserial modules. This can be stated as follows:

Definition 1

[6] A QTAG-module M is said to be a HT-module if every homomorphism \(f:M\longrightarrow M'\) is small whenever \(M'\) is a direct sum of uniserial modules.

Since these definitions are equivalent we are able to prove the following:

Theorem 5

Let M be a QTAG module with a submodule \(N\subseteq M\) such that \(H_k(M)\subseteq N\) for some \(k<\omega \). Then M is a HT-module if and only if N is a HT-module.

Proof

Suppose M is a HT-module and \(\displaystyle \frac{N}{K}\) is a direct sum of uniserial modules for some submodule K. Now \(\displaystyle {\frac{M/K}{N/K}}\simeq \displaystyle {\frac{M}{N}}\) is bounded as \(H_k(M)\subseteq N\), \(\displaystyle \frac{M}{K}\) is a direct sum of uniserial modules. Now there exists \(n<\omega \) such that \(Soc(H_n(M))\subseteq K\). Hence \(Soc(H_n(N))\subseteq K\), thus N is also a HT-module.

Conversely suppose N is a HT-module with \(\displaystyle {\frac{M}{T}}\) is direct sum of uniserial modules, for some \(T\subseteq M\). Now \(\displaystyle {\frac{N+T}{T}}\subseteq \displaystyle {\frac{M}{T}}\) is also a direct sum of uniserial modules. Therefore \(Soc(H_n(N+T))=Soc(H_n(N)+H_n(T))\subset T\) and \(Soc (H_n(N))\subseteq T\). Since \(H_k(M)\subseteq N\) we have \(Soc (H_{n+k}(M))\subset T\) implying that M is a HT-module. \(\square \)

Remark 1

Let two QTAG-modules M, \(M'\) be quasi-isomorphic. Then M is summable or \((\omega +n)\)-projective or a strong \(\omega \) -elongation of a totally projective module by a \((\omega +n)\) -projective module or HT-module, then so is \(M'\).

Proposition 1

Let M be a QTAG-module with a submodule N such that M/N is bounded. Then M is a direct sum of countably generated modules if and only if N is a direct sum of countably generated modules.

Proof

Suppose M is a diret sum of countably generated modules. Since M/N is bounded, there exists \(k<\omega \) such that \(H_k(M)\subseteq N\), therefore \(H_\omega (M)=H_\omega (N)\) is a direct sum of countably generated modules, therefore \(\displaystyle {\frac{N}{H_\omega (N)}}={\frac{N}{H_\omega (M)}}\subseteq \displaystyle {\frac{M}{H_\omega (M)}}\) is a direct sum of uniserial modules and N is also the direct sum of countably generated modules.

For the converse consider \(H_\omega (M)=H_\omega (N)\) which is a direct sum of countably generated modules and \(\displaystyle {\frac{N}{H_\omega (M)}}\) is a direct sum of uniserial modules. Now, \(\displaystyle {\frac{N}{H_\omega (M)}}=\bigcup \nolimits _{j<\omega }\displaystyle \left( \frac{N_j}{H_\omega (M)}\right) \) such that \(N_j\cap H_j(N)\subseteq H_\omega (M)\) for every j and \(N_j\cap H_{k+j}(M)\subseteq N_j\cap H_j(N)\subseteq H_\omega (M)\). Therefore the heights of the elements of \(\displaystyle \frac{N_j}{H_\omega (M)}\) are bounded, and by [8], M is a direct sum of countably generated modules. \(\square \)

Proposition 2

Let M be a QTAG module with a submodule N such that M/N is bounded. Then M is weakly \(\omega _1\) -separable if and only if N is weakly \(\omega _1\) -separable.

Proof

Since M/N is bounded there exists \(k<\omega \) such that \(H_k(M)\subseteq N\) and \(H_\omega (M)=H_\omega (N)\). Therefore M is separable if and only if N is separable. Since the property of being \(\omega _1\) -separable is inherited by the submodules, N is also weakly \(\omega _1\) -separable whenever M is weakly \(\omega _1\) -separable.

For the converse suppose N is weakly \(\omega _1\) -separable. Since \(H_k(M)\subseteq N\), \(H_k(M)\) is also weakly \(\omega _1\) -separable. Let K be a countably generated submodule of M. We have to show that \(g(\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M)))=\aleph _0\). If \(H_k(M)\cap K \) is finitely generated then it is a nice submodule in \(H_k(M)\) and \(\displaystyle \frac{H_k(M)}{K\cap H_k(M)}\simeq \displaystyle {\frac{H_k(M)+K}{K}}\) is separable because \(H_k(M)\) is separable. We also have \(\bigcap \nolimits _{i<\omega }\displaystyle \frac{(K+H_{k+i}(M))}{K}=\displaystyle \frac{\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M))}{K}=\bigcap \nolimits _{i<\omega }(H_i (H_k(M)+K)/K)=((H_k(M)+K)/K))^1=0\). Therefore \(\bigcap \nolimits _{i<\omega }(K+H_{k+i}(K))=K\). So we can write \(g(\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M)))=g(K)=\aleph _0\). Otherwise consider the case when \(g(H_k(M)\cap K)\) is \(\aleph _0\). Since N is weakly \(\omega _1\) -separable, so is \(H_k(M)\). Therefore \(\displaystyle {\left( \frac{H_k(M)}{H_k(M)\cap K}\right) }^1\simeq \displaystyle {\left( \frac{K+H_k(M)}{K}\right) }^1=\displaystyle \frac{\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M))}{K}\) and \(g\displaystyle {\left( \frac{\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M))}{K}\right) }\le \aleph _0\) implying that \(g(\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M)))=g(K)=\aleph _0\). Now in both cases we have \(g(\bigcap \nolimits _{i<\omega }(K+H_i(M))=g(\bigcap \nolimits _{i<\omega }(K+H_{k+i}(M))=\aleph _0\), therefore M is weakly \(\omega _1\) -separable as well.\(\square \)