A study of ${\omega }_1-(\omega +n)$ -projective QTAG-modules

Suppose K is an isotype submodule of a QTAG-module M. Then K is almost totally projective provided that K is separable in M.

Let K be an almost totally projective QTAG-module and suppose to the contrary that it is not separable in M. Then, there exists $m\in M$ such that, for each countably generated submodules T of K, we can find an element $k^{\ast }\in K$ such that $H(m+k^{\ast })>H(m+t)|$ for every $t\in T$ . Therefore we can find an ascending chain $$0=K_0\subseteq K_1\subseteq \dots \subseteq K_n\subseteq \dots$$ of countably generated submodules $K_n$ of K such that $K_n\in \mathcal{A}$ for each n and such that the following condition holds.

( $☆$ ) For every finite n, there exists $k_{n+1}\in K_{n+1}$ such that $H(m+k_{n+1})>H(m+k)$ for all $k\in K_n$ .

Now set $K_{\omega }={\bigcup }_{n<{\omega }_0}{K}_n$ and observe that $K_{\omega }$ is a countably generated submodule of K belonging to $\mathcal{A}$ . Since $K_{\omega }$ is countably generated, there exists $k^{\ast }\in K$ such that $H(m+t)<H(m+k^{\ast })$ for every $t\in K_{\omega }$ . Since $K_{\omega }$ is nice in K, there exists $k^{\prime }\in K_{\omega }$ such that $H_K(k^{\ast }-t)⩽H_K(k^{\ast }-k^{\prime })$ for all $t\in K_{\omega }$ . Moreover, since $k^{\prime }\in K_n$ for some n, there exists by condition ( $☆)$ an element $k^{\prime \prime }\in K_{n+1}$ such that $H(m+k^{\prime \prime })<H(m+k^{\ast })$ . So, we now have $H_K(k^{\ast }-k\prime \prime )⩽H_K(k^{\ast }-k^{\prime })$ . Since K is isotype in M, we have $H(m+k\prime )<H((m+k^{\ast })-(m+k\prime \prime ))=H(k^{\ast }-k^{\prime \prime })=H_K(k^{\ast }-k^{\prime \prime })⩽H_K(k^{\ast }-k\prime )=H(k^{\ast }-k\prime )=H(m+k\prime )$ which leads to a contradiction and proving our desired result.

If $H_{\sigma }(M)$ and $M/H_{\sigma }(M)$ are almost totally projective, for any ordinal $\sigma$ then M is also almost totally projective.

We know that a submodule K is a nice submodule of M if and only if $H_{\sigma }(K)$ is a nice submodule of $H_{\sigma }(M)$ and $K+H_{\sigma }(M)/H_{\sigma }(M)$ is a nice submodule of $M/H_{\sigma }(M)$ . Hence, the properties satisfying the three conditions for a family of nice submodules to be almost totally projective, for $H_{\sigma }(M)$ and $M/H_{\sigma }(M)$ lead to satisfying the same conditions for the module M.

The arbitrary direct sums of almost totally projective QTAG-modules are almost totally projective.

Suppose $M=A_{i\in I}{M}_i$ . As it is well known, if $K=A_{i\in I}{K}_i$ with $K_i\subseteq M_i$ for all $i\in I$ , then K is a nice submodule of M if and only if $K_i$ is a nice submodule of $M_i$ for each $i\in I$ . Hence, the three properties of almost totally projectives modules satisfying by $M_i$ ensures that M will certainly satisfies the same and hence the result follows.

“The separable $\mathit{QTAG}$ -module M is said to be almost direct sum of uniserial modules if it possesses a collection $\mathcal{N}$ consisting of nice submodules of M which satisfies the following three conditions:

1. $\{0\}\in \mathcal{N}$

2. $\mathcal{N}$ is closed with respect to ascending unions, i.e., if $N_i\in \mathcal{N}$ with $N_i\subseteq N_j$ whenever $i⩽j(i,j\in I)$ then ${\cup }_{i\in I}{N}_i\in \mathcal{N}$ ;

3. If K is a countably generated submodule of M, then there is $L\in \mathcal{N}$ (that is, a nice submodule L of M) such that $K\subseteq L$ and L is countably generated.

Definition 2 Hasan, 2018

The QTAG-module M is said to be almost $(\omega +n)$ -projective if there exists $N\subseteq H^n(M)$ such that $M/N$ is almost direct sum of uniserial modules.

The module M is almost $(\omega +n)$ -projective if and only if $M\cong S/T$ for some almost direct sum of uniserial module S and $T\subseteq H^n(S)$ .

“ $\Rightarrow$ . Let N be a module with $H_n(N)=M$ and let $S=N/L$ where $L\subseteq H^n(M)$ such that $M/L$ is almost direct sum of uniserial modules. Consequently, $H_n(S)=H_n(N/L)=M/L$ is almost direct sum of uniserial module and hence, by Proposition 2, S is almost direct sum of uniserial module too. Supposing $T=H^n(N)/L$ , we deduce that $S/T\cong N/H^n(N)\cong H_n(N)=M$ , as required. $\mathrm{“}\Leftarrow$ . Assume that $M\cong S/T$ , which we without loss of generality interpret as an equality, whence we get $L=H^n(S)/T\subseteq H^n(M)$ . Thus, again in view of Proposition 2, $M/L\cong S/H^n(S)\cong H_n(S)$ is almost direct sum of uniserial module, as desired.

A QTAG-module M is called almost ${\omega }_1-(\omega +n)$ -projective if there is a countably generated nice submodule N such that $M/N$ is almost $(\omega +n)$ -projective.

A submodule of an almost $(\omega +n)$ -projective module is also almost $(\omega +n)$ -projective.

. Let $L\subseteq H^n(M)$ such that $M/L$ is almost $\mathrm{\Sigma }$ -uniserial i.e, $M/L$ is almost direct sum of uniserial modules and suppose that $T\subseteq M$ . Then $(T+L)/L\subseteq M/L$ is again almost $\mathrm{\Sigma }$ -uniserial, and $T/(T\cap L)\cong (T+L)/L$ with $T\cap L\subseteq H^n(T)$ , as desired.

The submodule of an almost ${\omega }_1-(\omega +n)$ -projective module is almost ${\omega }_1-(\omega +n)$ -projective as well.

Assume that $P\subseteq M$ where M is almost ${\omega }_1-(\omega +n)$ -projective. Thus $M/N$ is almost $(\omega +n)$ -projective for some countably generated submodule N. Moreover, $(N+P)/P\subseteq M/P$ is almost $(\omega +n)$ -projective too by Proposition 5, and $P/(P\cap N)\cong (N+P)/P$ . Since $N\cap P\subseteq N$ is countably generated, we are done.

Let M be a separable module with a countably generated nice submodule N. Then M is almost direct sum of uniserial modules if and only if $M/N$ is almost direct sum of uniserial modules.

Let N be a countably generated submodule of a QTAG module M with $M/N$ as almost direct sum of uniserial submodules. Then M is the sum of a countably generated submodule and an almost direct sum of uniserial modules.

Since $H_{\omega }(M)\subseteq N$ is countably generated, we may isomorphically embed it in an essential submodule of $M/L$ where L is a high submodule of M and thus $M/L$ will also be countably generated. In fact, $H_{\omega }(M)\cong \left(H_{\omega }(M)\oplus L\right)/L\subseteq M/L$ where it is easily checked that $\left(H_{\omega }(M)\oplus L\right)/L$ is essential in $M/L$ because L is maximal with respect to intersecting $H_{\omega }(M)$ trivially. This provide evidence to support our claim. Furthermore, one can write that $M=L+S$ for some countably generated submodule S. We know that by Begam, 2014, If $M/N$ is almost direct sum of uniserial submodules for some QTAG module M and its countably generated submodule N, then M is almost simply presented and a high submodule of an almost simply presented module is almost direct sum of uniserial module, we obtain that M must be almost direct sum of uniserial module and so the required decomposition.

Suppose that T is an almost direct sum of uniserial module and S is its countably generated submodule. Then $T/S$ is the sum of a countably generated module and an almost direct sum of uniserial module.

Using Remark 1, one can infer, for any countably generated nice submodule C of T as $S\subseteq C$ that $T/C$ remain almost direct sum of uniserial module. But $T/C\cong (T/S)/(C/S)$ , where $C/S$ is countably generated. Now use of Lemma 1 ensures the desired decomposition of $T/S$ .

A separable QTAG module M is almost $(\omega +n)$ -projective if and only if $M/N$ is almost $(\omega +n)$ -projective, where N be a countably generated nice submodule of the sepearable module M.

“ $\Rightarrow$ Suppose that $M/L$ is almost direct sum of uniserial module, for some $L\subseteq H^n(M)$ . Observe that the two isomorphisms hold: $$(M/N)/(L+N)/N\cong M/(L+N)\cong (M/L)/(L+N)/L$$

Since $(L+N)/L\cong N/(L\cap N)$ is countably generated, it follows from Begam, 2014 that $T/D$ is almost simply presented where we put $M/N=T$ and $(L+N)/N=D$ . Therefore, $$T/\left[{\cap }_{n<\omega }\left(H_n(T)+D\right)\right]\cong (T/D)/[{\cap }_{n<\omega }(H_n(T)+D)]/D=(T/D)/H_{\omega }(T/D)$$ is almost direct sum of uniserial modules with $H_t\left({\cap }_{n<\omega }\left(H_n(T)+D\right)\right)=H_{\omega }(T)=H_{\omega }(M/N)=(H_{\omega }(M)+N)/N=\{0\}$ . Thus by Definition 2 the quotient module $T=M/N$ is almost $H_{\omega +n}$ -projective, as claimed.

$\mathrm{“}\Leftarrow$ . For the reverse implication suppose that $M/N$ be almost $H_{\omega +n}$ -projective. Therefore, there is a quotient $S/N$ with $S\subseteq M$ and $H_n(S)\subseteq N$ such that $(M/N)/(S/N)\cong$ $M/S$ is almost direct sum of uniserial modules. Hence S is the direct sum of a countably generated and an $n$ -bounded module, say $S=A\oplus Q$ where A is countably generated and $H_n(Q)=\{0\}$ . We further infer that $$M/S=M/(A\oplus Q)\cong (M/Q)/(A\oplus Q)/Q$$ where $(A\oplus Q)/Q\cong A$ is countably generated and so $M/Q$ is almost simply presented. Therefore, $(M/Q)/H_{\omega }(M/Q)\cong$ $M/\left[{\cap }_{n<\omega }\left(H_n(M)+Q\right)\right]$ is almost direct sum of uniserial modules with $H_t\left({\cap }_{n<\omega }\left(H_n(M)+Q\right)\right)=H_{\omega }(M)=0$ . Finally, M is almost $H_{\omega +n}$ -projective, as expected.

Suppose that M is a separable almost ${\omega }_1-(\omega +n)$ -projective module. Then M is almost $(\omega +n)$ -projective.

Let N be a countably generated submodule of the module M such that $M/N$ is almost $(\omega +n)$ -projective. Application of Proposition 7 gives that M is almost $(\omega +n)$ -projective, as asserted.

(a) If M is almost $(\omega +n)$ -projective, then $M/H_{\gamma }(M)$ is almost $(\omega +n)$ -projective for any ordinal $\gamma$ .

(b) If M is (nicely) almost ${\omega }_1-(\omega +n)$ -projective, then $M/H_{\gamma }(M)$ is (nicely) almost ${\omega }_1-(\omega +n)$ -projective for all ordinals $\gamma$ .

The above statements are trivial for the ordinals $\gamma <\omega$ . So, we will discuss the case $\gamma ⩾\omega$ .

(a) Suppose $M/L$ is almost direct sum of uniserial modules for some $L\subseteq H^n(M)$ . So, $H_{\omega }(M)\subseteq$ L and hence $H_{\gamma }(M)\subseteq L$ for each ordinal $\gamma ⩾\omega$ . Furthermore, $M/L\cong$ $\left(M/H_{\gamma }(M)\right)/\left(L/H_{\gamma }(M)\right)$ is almost direct sum of uniserial modules with $L/H_{\gamma }(M)\subseteq H^n(\left(M/H_{\gamma }(M)\right))$ , as desired.

(b) First, we will discuss the case for nice submodules. So if $M/N$ is almost $(\omega +n)$ -projective for some countably generated nice submodule N, we deduce with the help of part (a), that $$\begin{array}{cc}\hfill & (M/N)/H_{\gamma }(M/N)=(M/N)/\left(H_{\gamma }(M)+N\right)/N\cong \hfill \\ \hfill & M/(H_{\gamma }(M)+N)\cong \left(M/H_{\gamma }(M)\right)/\left(H_{\gamma }(M)+N\right)/p^{\lambda }G\hfill \end{array}$$ is almost $(\omega +n)$ -projective. Since $\left(H_{\gamma }(M)+N\right)/H_{\gamma }(M)\cong N/\left(H_{\gamma }(M)\cap N\right)$ is obviously countably generated and nice in $M/H_{\gamma }(M)$ , Fuchs (1970 and 1973.), we are done.

Now we will discuss the part without $\mathrm{“}\mathit{niceness}$ . To show it, suppose $M/N$ be almost $(\omega +n)$ -projective for some countably generated submodule N. We claim that $$(M/N)/\left(H_{\gamma }(M)+N\right)/N\cong M/\left(H_{\gamma }(M)+N\right)\cong \left(M/H_{\gamma }(M)\right)/\left(H_{\gamma }(M)+N\right)/H_{\gamma }(M)\text{.}$$ is almost $(\omega +n)$ -projective. In fact, if P is an almost $(\omega +n)$ -projective module with $T\subseteq H_{\gamma }(P)$ , then $P/T$ is also almost $(\omega +n)$ -projective. To this goal, write $P/Q$ is almost direct sum of uniserial modules for some $Q\subseteq H^n(P)$ . Thus $H_{\omega }P\subseteq Q$ whence $T\subseteq H_{\gamma }(P)\subseteq H_{\omega }(P)\subseteq Q$ . This gives that $P/Q\cong$ $(P/T)/(Q/T)$ is almost direct sum of uniserial modules for $Q/T\subseteq H^n(P/T)$ and means that $P/T$ is really as desired. We just apply this assertion to $P=M/N$ and $T=\left(H_{\gamma }(M)+N\right)/N\subseteq H_{\gamma }(M/N)=H_{\gamma }(P)$ and the claim is established. Furthermore, by what we have previously shown, $(M/H_{\gamma }(M))/(H_{\gamma }(M)+N)/H_{\gamma }(M)$ being almost $(\omega +n)$ -projective with countably generated $(H_{\gamma }(M)+N)/H_{\gamma }(M)\cong N/\left(N\cap H_{\gamma }(M)\right)$ ensures that $M/H_{\gamma }(M)$ is almost ${\omega }_1-(\omega +n)$ -projective, as formulated.

If M is almost ${\omega }_1-(\omega +n)$ -projective, then $M/H_{\omega }(M)$ is almost $(\omega +n)$ -projective.

Suppose M is a module such that $H_{\omega }(M)$ is countably generated. Then M is almost ${\omega }_1-(\omega +n)$ -projective if and only if $M/H_{\omega }(M)$ is almost $(\omega +n)$ - projective.

The direct part follows trivially by Definition 3 while the reverse implication part can be obtained using Corollary 3.

The module M is (nicely) almost ${\omega }_1-(\omega +n)$ -projective if and only if $H_{\omega +n}(M)$ is countably generated and $M/H_{\omega +n}(M)$ is (nicely) almost ${\omega }_1-(\omega +n)$ projective.

The direct implications follows from Proposition 8 (b) by substituting $\gamma =\omega +n$ . As for the other way round, suppose $\left(M/H_{\omega +n}(M)\right)/\left(N/H_{\omega +n}(M)\right)\cong M/N$ is almost $(\omega +n)$ -projective for some countably generated (nice) quotient $N/H_{\omega +n}(M)$ such that $N\subseteq M$ . But N is countably generated (and nice) in M, so that M is (nicely) almost ${\omega }_1-(\omega +n)$ -projective, as claimed.

Suppose that $H_{\alpha }(M)$ is countably generated for some ordinal $\alpha$ . Then M is almost ${\omega }_1-(\omega +n)$ -projective if and only if $M/H_{\alpha }(M)$ is almost ${\omega }_1-(\omega +n)$ -projective.

The direct implication can be obtained using Proposition 8 while the reverse implication follows on the same line of the proof of Theorem 2.

The direct sums of almost $(\omega +n)$ -projective modules are almost $(\omega +n)$ -projective modules.

Suppose $M=\underset{i\in I}{A}{M}_i$ where all components $M_i$ are almost $(\omega +n)$ - projective. So, $M_i/N_i$ ’s are almost direct sum of uniserial modules for some $N_i\subseteq H^n(M_i)$ . Furthermore, putting $N=\underset{i\in I}{A}{N}_i$ , we infer that $N\subseteq H^n(M)$ and that $M/N\cong \underset{i\in I}{A}{M}_i/N_i$ are almost direct sums of uniserial modules owing to Mehdi et al. (2006), as expected.

Suppose $M=\underset{t\in T}{A}{M}_t$ is a module for some index set T. Then M is almost ${\omega }_1-(\omega +n)$ -projective if and only if $M_t$ is almost ${\omega }_1-(\omega +n)$ -projective for each index $t\in T$ , and there exists a countable subset $B\subseteq T$ such that $M_t$ ’s are almost $(\omega +n)$ -projective for all $t\in T⧹B$ .

For the direct implication, suppose that X be a countably generated submodule of M such that $M/X$ is almost $(\omega +n)$ -projective. From Proposition 6 it follows that all $M_t$ ’s are almost ${\omega }_1-(\omega +n)$ -projective. Clearly, $X\subseteq {\oplus }_{t\in B}{M}_t$ for some $B\subseteq T$ with $|B|⩽{\aleph }_0$ . Therefore,

$M/X\cong \left[\left(\underset{t\in B}{A}{M}_t\right)/X\right]\oplus \left[\underset{t\in T⧹B}{A}{M}_t\right]$ , so that $\underset{t\in T⧹B}{A}{M}_t$ , and hence $M_t$ ’s are almost $(\omega +n)$ -projective for every $t\in T⧹B$ in conjunction with Proposition 5.

For the reverse implication suppose all factors $M_t/X_t$ be almost $(\omega +n)$ -projective for some countably generated submodules $X_t\subseteq M_t$ . Set $X={\oplus }_{t\in B}{X}_t$ , whence X is countably generated. However, $$M/X\cong \left[{\oplus }_{t\in B}\left(M_t/X_t\right)\right]\oplus \left[{\oplus }_{t\in T⧹B}{M}_t\right],$$ and so by Proposition 9 we conclude that $M/X$ is almost $(\omega +n)$ -projective, as required.

The countable direct sum of almost ${\omega }_1-(\omega +n)$ -projective module is an almost ${\omega }_1-(\omega +n)$ - projective module.

The following conditions are equivalent:

1. M is almost ${\omega }_1-(\omega +n)$ -projective;

2. $M/N$ is the sum of a countably generated module and an almost direct sum of uniserial module where $H_n(N)=\{0\}$ ;

3. $M\cong \mathfrak{B}/S$ where $\mathfrak{B}$ is the sum of a countably generated module and an almost direct sum of uniserial module and $H_n(S)=\{0\}$ ;

4. $M/T$ is almost direct sum of uniserial module, where $H_n(T)$ is countably generated $(T$ is the direct sum of a countably generated module and an n-bounded module);

5. $M\cong W/H$ , where W is almost direct sum of uniserial module and $H_n(B)$ is countably generated $(B$ is the direct sum of a countably generated module and a n-bounded module);

6. $M\cong X/Y$ , where X is almost $(\omega +n)$ -projective and Y is countably generated;

7. $M/U$ is countably generated, where U is almost $(\omega +n)$ -projective.

We start with $\mathrm{“}(1)\Rightarrow (7)$ : Suppose $M/Q$ is almost $(\omega +n)$ -projective for some countably generated submodule $Q\subseteq M$ . Let $U\subseteq M$ be maximal with respect to $U\cap Q=\{0\}$ .

Clearly $U\cong (U\oplus Q)/Q\subseteq M/Q$ , so that Proposition 5 applies to get that Uis almost $(\omega +n)$ -projective too.

On the other way round, $Q\cong (Q\oplus U)/U$ where the latter is an essential submodule of $M/U$ , and thus $M/U$ will be countably generated. In fact, for any $A\subseteq M$ with $A\ne U$ we obtain by the modular law from Fuchs (1970 and 1973.) that $[(Q\oplus$ $U)/U]\cap (A/U)=((Q\oplus U)\cap A)/U=(U+Q\cap A)/U\ne \{0\}$ because $Q\cap A⊈U$ since $Q\cap A\ne \{0\}$ . Thus (7) follows.

$(1)\iff (4)$ . Suppose that $M/T$ is almost direct sum of uniserial modules, where $T=Q\oplus N$ for some countably generated submodule Q and $n$ -bounded submodule N. But $M/T=M/(Q\oplus N)\cong M/Q/(Q\oplus N)/Q$ , and $(Q\oplus N)/Q\cong N$ is $n$ -bounded. Therefore, $M/Q$ is almost $(\omega +n)$ -projective and (1) holds. Conversely suppose that $M/Q$ be almost $(\omega +n)$ -projective for some countably generated submodule Q. Hence there is an $n$ -bounded submodule $T/Q$ with $T\subseteq M$ such that $(M/Q)/(T/Q)\cong M/T$ is almost direct sum of uniserial modules. Since $H_n(T)\subseteq Q$ is countably generated, we are done.

$(2)\iff (4)$ . First, we note the following helpful fact: Setting $X=B+W$ , where B is countably generated and W is almost direct sum of uniserial modules, there exists a countably generated module C such that $X/C$ is almost direct sum of uniserial modules. Indeed, $B\cap W$ being a countably generated submodule of W forces that $B\cap W\subseteq F$ where F is a countably generated nice submodule of W whence by Remark 1 we infer that $W/F$ is almost direct sum of uniserial module. It therefore follows that $X/F=[(B+F)/F]\oplus [W/F]$ . That is why, $(X/F)/(B+F)/F\cong X/(B+F)\cong W/F$ is almost direct sum of uniserial modules. Denoting $C=B+F$ , we are done. Furthermore, applying the last observation to $M/N$ we obtain that $(M/N)/(T/N)\cong M/T$ is almost direct sum of uniserial modules, where $T/N$ is countably generated. Hence $H_n(T)$ is countably generated, as stated. For the reverse part suppose that $M/T$ is almost direct sum of uniserial modules with $T=\mathfrak{B}\oplus N$ where $\mathfrak{B}$ is countably generated and N is bounded by $n$ . However, $M/T\cong (M/N)/(T/N)$ is almost direct sum of uniserial modules with countably generated $T/N\cong \mathfrak{B}$ , so that Lemma 1 is applicable for $M/N$ to finish the equivalence.

$(6)\iff (5)$ . First, assume that $M\cong X/Y$ for some almost $(\omega +n)$ - projective module X and its countably generated submodule Y. Using Proposition 4, one may write that $X=W/N$ where W is almost direct sum of uniserial modules with $H_n(N)=\{0\}$ , and $Y=D/N$ is countably generated with $D\subseteq W$ . Furthermore, $M\cong W/D$ and since $D=N+C$ for some countably generated submodule C, one may derive that $H_n(D)=H_n(C)$ is countably generated, as required.

For the reverse implication, let us assume that $M\cong W/D$ where W is almost direct sum of uniserial modules and $H_n(D)$ is countably generated. Since D is the direct sum of a countably generated submodule B and an $H_n$ -bounded module V, say $D=B\oplus V$ , one may deduce that $M\cong W/(B\oplus V)\cong (W/V)/(B\oplus V)/V$ . However, using Proposition 4, $W/V$ is almost $(\omega +n)$ -projective, whereas $(B\oplus V)/V\cong B$ is countably generated. This ensures that (6) holds, thus completing the verification of the desired equivalence.

$(7)\Rightarrow (6)$ . Suppose that $M/U$ is countably generated for some almost $(\omega +n)$ - projective submodule U. Let A be a countably generated submodule which is the direct sum of uniserial modules and $\beta :A\to M$ be a homomorphism such that $M=U+\beta (A)$ . If we set $F=U\oplus A$ , then F is almost $(\omega +n)$ -projective appealing to Proposition 9. If now we let $g:U\to M$ be the identity map, then we have a surjective homomorphism $\alpha :F\to M$ . If B is its kernel, then obviously $B\cap U=\{0\}$ ; in fact, $x\in B\cap U$ forces that $\alpha (x)=x=0$ . Hence $M\cong F/B$ and B is isomorphic to a submodule of A. Thus B is countably generated, and we are done.

$(5)\Rightarrow (3)$ . Let $M\cong W/D$ where W is almost direct sum of uniserial modules and $D=G\oplus N$ where G is countably generated and N is bounded by $n$ . Since $M\cong (W/G)/(D/G)$ and $D/G\cong N$ is $n$ -bounded, we just take into account Corollary 1 to conclude that $W/G$ is the sum of a countably generated submodule and an almost direct sum of uniserial modules, as desired.

$(3)\Rightarrow (2)$ . Suppose $M=\mathfrak{B}/S$ where $\mathfrak{B}=B+W$ with countably generated submodule B and almost direct sum of uniserial module W. Since $S\subseteq H^n(\mathfrak{B})$ , set $N=H^n(\mathfrak{B})/S\subseteq M$ . Thus $M/N\cong \mathfrak{B}/H^n(\mathfrak{B})\cong H_n(\mathfrak{B})=H_n(B)+H_n(W)S$ remains again the sum of a countably generated submodule and an almost direct sum of uniserial modules.

$(3)\Rightarrow (7)$ . Let us express $M=\mathfrak{B}/S$ , where $\mathfrak{B}=B+W$ and B is countably generated whereas W is almost direct sum of uniserial modules. But $\mathfrak{B}/S=(C+S)/S+(W+S)/S$ . Observing that $(C+S)/S\cong C/(C\cap S)$ is countably generated and $(W+S)/S\cong S/(S\cap V)$ is almost $(\omega +n)$ -projective in conjunction with Proposition 4, we routinely see that $M/U$ is countably generated for $U=(W+S)/S$ , as needed.

(2) $\iff (7)$ . Suppose first that $M/U$ is countably generated for some almost $(\omega +n)$ -projective module U. So, we write $M=U+C$ for some countably generated submodule $C\subseteq M$ . Consequently, $U/N$ is almost direct sum of uniserial module for some $N\subseteq$ $H^n(U)$ and thus $M/N=[U/N]+[(C+N)/N]$ where $(C+N)/N\cong C/(C\cap N)$ is countably generated. Thus (2) is satisfied. Conversely, let $M/N=(X/N)+(Y/N)$ where the first summand is countably generated while the second is almost direct sum of uniserial modules. So, $M=X+U$ and $X=N+G$ where G is countably generated, which gives $M=U+G$ . But U is almost direct sum of uniserial modules and $M/U=(G+U)/U\cong G/(G\cap U)$ is countably generated. Hence (7) is fulfilled.

Let N be a submodule of M such that $M/N$ is countaly generated. Then M is almost ${\omega }_1-(\omega +n)$ -projective if and only if N is almost ${\omega }_1-(\omega +n)$ projective.

The direct part follows by using Proposition 6. For the reverse implication, according to point (7) of Theorem 3, we write that $N/T$ is countably generated for some almost $(\omega +n)$ -projective module T. Therefore, $M/N\cong (M/T)/(N/T)$ being countably generated implies that $M/T$ is countably generated and again point (7) of Theorem 3 yields that M is almost ${\omega }_1-(\omega +n)$ -projective, as claimed.

Let L be a countably generated submodule of M. Then M is almost ${\omega }_1-(\omega +n)$ -projective if and only if $M/L$ is almost ${\omega }_1-(\omega +n)$ projective.

$\mathrm{“}\Leftarrow$ : Suppose $M/L$ is almost ${\omega }_1-(\omega +n)$ -projective for the countably generated submodule L. Therefore there is a countably generated submodule $P/L$ of $M/L$ with $P\subseteq M$ such that $(M/L)/(P/L)\cong M/P$ is almost $(\omega +n)$ -projective. Since P remains countably generated, M must be almost ${\omega }_1-(\omega +n)$ -projective, as claimed.

$\mathrm{“}\Rightarrow$ : Suppose $M/Y$ be almost $(\omega +n)$ -projective for some countably generated submodule Y. By part (6) of Theorem 3, we have that $(M/Y)/(L+Y)/Y\cong M/(L+Y)\cong (M/L)/(L+Y)/L$ is almost $(\omega +n)$ -projective because $(L+Y)/Y\cong L/(L\cap Y)$ is countably generated. Since $(L+Y)/L\cong Y/(Y\cap L)$ remains countably generated, an appeal to the above part proved assures that $M/L$ is almost $(\omega +n)$ -projective, as asserted.

The class of almost ${\omega }_1-(\omega +n)$ -projective modules is closed under ${\omega }_1$ -bijections, and is the minimal class containing almost $(\omega +n)$ -projectives with that property.

The first part follows by a combination of Theorems 4 and 5 along with (Lemma 2.9, Sikander, 2019). The remaining part can be proved easily by using Theorem 3 and (Proposition 2.10, Sikander, 2019)

The module M is almost ${\omega }_1-(\omega +n)$ -projective if and only if there exists a countably generated nice submodule K which satisfies the inequalities $H_{\omega +n}(M)\subseteq K\subseteq H_{\omega }(M)$ such that $M/K$ is almost $(\omega +n)$ -projective.

In view of part (3) of Theorem 3, we write $M=L/P$ where L is the sum of a countably generated module B and an almost direct sum of uniserial modules Q, say $L=B+Q$ , and $P\subseteq H^n(L)$ . Setting $K=(H_{\omega }(L)+P)/P$ , we observe that $$M/K\cong L/(H_{\omega }(L)+P)\cong (L/H_{\omega }(L))/(H_{\omega }(L)+P)/H_{\omega }(L)$$

We now prove two things about L, that are, $L/H_{\omega }(L)$ is almost direct sum of uniserial modules, and $H_{\omega }(L)$ is countably generated whence so is $H_{\omega }(L)/(H_{\omega }(L)\cap P)\cong K$ . In fact, since $B\cap Q\subseteq Q$ is countably generated, there is a countably generated nice submodule $\mathfrak{B}$ of Q such that $\mathfrak{B}\supseteq B\cap Q$ . That is why, $L/\mathfrak{B}=[(B+\mathfrak{B})/\mathfrak{B}]\oplus [\mathfrak{B}/R]$ . However, $(B+\mathfrak{B})/\mathfrak{B}$ remains countably generated, whereas $Q/\mathfrak{B}$ remains separable. Consequently, $H_{\omega }(L/\mathfrak{B})=H_{\omega }$ is countably generated and hence the same holds for its submodule $(H_{\omega }(L)+\mathfrak{B})/\mathfrak{B}\cong H_{\omega }(L)/\left(\mathfrak{B}\cap H_{\omega }(L)\right)$ . But $(\mathfrak{B}\cap H_{\omega }(L))$ is countably generated, so that $H_{\omega }(L)$ is countably generated, indeed.

For the other claim by using Remark 1, we have that $(L/\mathfrak{B})/(B+\mathfrak{B})/\mathfrak{B}\cong L/(B+\mathfrak{B})\cong Q/\mathfrak{B}$ is almost direct sum of uniserial modules. Since $B+\mathfrak{B}$ remains countably generated Then $L/H_{\omega }(L)$ is really almost direct sum of uniserial modules Begam, 2014. Furthermore, since $(H_{\omega }(L)+P)/H_{\omega }(L)\cong P/\left(P\cap H_{\omega }(L)\right)$ is bounded by $n$ , it now follows from Proposition 4 that $M/K$ must be almost $(\omega +n)$ - projective, as desired.

Finally, it is routinely seen that $K\subseteq H_{\omega }(M)$ and that $K\supseteq H_{(}\omega +n)(M)$ because $(H_{\omega +n}(M)+K)/K\subseteq H_{\omega +n}(M/K)=\{0\}$ . It is next easily checked that such a submodule K satisfying the above two inequalities should be nice in M, as desired.