A new version of a theorem of Kaplansky

TL;DR

This paper extends Kaplansky's theorem by proving a $w$-version involving hereditary torsion theories, broadening the understanding of projective modules over commutative rings.

Contribution

It introduces and proves a new $w$-version of Kaplansky's theorem for projective modules within hereditary torsion theories.

Findings

Established the $w$-version of Kaplansky's theorem

Extended the class of modules where Kaplansky's decomposition applies

Provided new insights into projective modules over commutative rings

Abstract

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the $w$-version of this theorem, where $w$ is a hereditary torsion theory for modules over a commutative ring.

Equations

$$\phi :R\to J^{\ast }:={\mathrm{Hom}}_R(J,R)$$

$${\rm tor_{\rm GV}}(M):=\{x\in M~{}|~{}\mbox{$Jx=0$ for some $J\in{\rm GV}(R)$}\}.$$

$$M_{w}:=\{x\in E(M)~{}|~{}\mbox{$Jx\subseteq M$ for some $J\in{\rm GV}(R)$}\}$$

$$\mathrm{to}{\mathrm{r}}_{\mathrm{GV}}=\text{mbox}(\{GV-torsionmodules\},\{GV-torsionfreemodules\})$$

$$0\to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to 0$$

$$0\to K\to P\to M\to 0$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$h^{\prime }(c)=\beta (b)-f^{\prime }h(b).$$

$$\beta (b-b_1)-f^{\prime }h(b-b_1)=\beta f(a)-f^{\prime }hf(a)=\beta f(a)-f^{\prime }\alpha (a)=0,$$

$$g^{\prime }\beta (b)-g^{\prime }{h}^{\prime }g(b)=\gamma g(b)-\gamma g(b)=0,$$

$$f^{\prime }hf(a)=f^{\prime }(a^{\prime })=\beta f(a)-h^{\prime }gf(a)=f^{\prime }\alpha (a).$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$g_{\ast }:{\mathrm{Hom}}_R(M,B)\to {\mathrm{Hom}}_R(M,C)$$

$$g_{\ast }^{\prime }:{\mathrm{Hom}}_R(M,F)\to {\mathrm{Hom}}_R(M,M)$$

$$0\to {\mathrm{Hom}}_R(M,N)\to {\mathrm{Hom}}_R(F,N)\stackrel{f^{\ast }}{⟶}{\mathrm{Hom}}_R(L,N)\to {\mathrm{Ext}}_R^1(M,N)\to 0$$

$${\mathrm{Hom}}_R(M,B)\stackrel{g_{\ast }}{⟶}{\mathrm{Hom}}_R(M,C)\to {\mathrm{Ext}}_R^1(M,\ker (g))$$

$$g{h}_k(y)=g\left(\sum _i{f}_{k_i}(y)e_i\right)=\sum _i{f}_{k_i}(y)x_i=d_{k}y={\eta }_{d_k}^{M^{\prime }}(y),$$

$$\mbox{$M=\bigoplus\{R/J~{}|~{}J\in{\rm GV}(R)\}$}.$$

$${\mathfrak{m}}^{\ast }={\mathrm{Hom}}_R(\mathfrak{m},R)\cong {\mathfrak{m}}^{-1}=R$$

$$J_0\subseteq \bigcap _{n=1}^{\infty }{\mathfrak{m}}^n$$

$${\mathrm{Hom}}_R(M,P)\stackrel{g_{\ast }}{⟶}{\mathrm{Hom}}_R(M,M)\to {\mathrm{Ext}}_R^1(M,K)$$

$${\mathrm{Hom}}_R(M_w,N)\stackrel{{\mu }^{\ast }}{⟶}{\mathrm{Hom}}_R(M,N)\to {\mathrm{Ext}}_R^1(T,N)\to {\mathrm{Ext}}_R^1(M_w,N).$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mbox{$g_{1}=g|_{F_{1}}$, $M_{1}={\rm im}(g_{1})$, $P_{1}=f^{-1}({\rm ker}(g_{1}))$ and $f_{1}=f|_{P_{1}}$}.$$

$$F(H)=0$$

$$F(H)=\underset{j\in H}{⨁}{F}_j$$

$$P(H)=f^{-1}(\ker (g_H))$$

Full text

Key Words: $w$-split modules; $w$-projective modules; Kaplansky’s theorem on projective modules

$2010$ Mathematics Subject Classification: 13C13; 13D07; 13D30

$\sharp$ Address correspondence to Dr. Lei Qiao, School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610066, China; E-mail: lqiao@sicnu.edu.cn

1. Introduction

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules (see [7]). This is equivalent to saying that every projective module can be filtered by countably generated and projective modules. In [10], Šťovíček and Trlifaj applied Hill’s method [5] to extend Kaplansky’s theorem on projective modules to the setting of cotorsion pairs. Later, Enochs et al. [1] also got the analogous version of Kaplansky’s Theorem for cotorsion pairs for a more general setting on concrete Grothendieck categories. Moreover, several versions of Kaplansky’s theorem have been discussed in the literature. For example, a categorical version of Kaplansky’s theorem on projective modules is proved in [8, Lemma 3.8] by Osofsky. Also, in [11], Estrada et al. prove a version of Kaplansky’s Theorem for quasi-coherent sheaves, by using Drinfeld’s notion of almost projective module and the Hill Lemma.

The purpose of this article is to present a $w$-version of Kaplansky’s theorem on projective modules, where $w$ is a hereditary torsion theory for modules over a commutative ring. Next, we shall review some terminology related to the hereditary torsion theory $w$, see [12] for details. Throughout, $R$ denotes a commutative ring with an identity element and all modules are unitary.

Recall from [15] that an ideal $J$ of $R$ is called a Glaz-Vasconcelos ideal (a ${\rm GV}$-ideal for short) if $J$ is finitely generated and the natural homomorphism

[TABLE]

is an isomorphism. Notice that the set ${\rm GV}(R)$ of GV-ideals of $R$ is a multiplicative system of ideals of $R$. Let $M$ be an $R$-module. Define

[TABLE]

Thus ${\rm tor_{\rm GV}}(M)$ is a submodule of $M$. Now $M$ is said to be ${\rm GV}$-torsion (resp., ${\rm GV}$-torsionfree) if ${\rm tor_{\rm GV}}(M)=M$ (resp., ${\rm tor_{\rm GV}}(M)=0$). A GV-torsionfree module $M$ is called a $w$-module if ${\rm Ext}^{1}_{R}(R/J,M)=0$ for all $J\in{\rm GV}(R)$. Then projective modules and reflexive modules are both $w$-modules. In [12, Theorem 6.7.24], it is shown that all flat modules are $w$-modules. Also it is known that a GV-torsionfree $R$-module $M$ is a $w$-module if and only if ${\rm Ext}^{1}_{R}(N,M)=0$ for every GV-torsion $R$-module $N$ (see [12, Theorem 6.2.7]). For any GV-torsionfree module $M$,

[TABLE]

is a $w$-submodule of $E(M)$ containing $M$ and is called the $w$-envelope of $M$, where $E(M)$ denotes the injective envelope of $M$. It is clear that a GV-torsionfree module $M$ is a $w$-module if and only if $M_{w}=M$.

It is worthwhile to point out that from a torsion-theoretic point of view, the notion of $w$-modules coincides with that of ${\rm tor_{\rm GV}}$-closed (i.e., ${\rm tor_{\rm GV}}$-torsionfree and ${\rm tor_{\rm GV}}$-injective) modules, where ${\rm tor_{\rm GV}}$ is the torsion theory whose torsion modules are the GV-torsion modules and torsionfree modules are the GV-torsionfree modules, that is, the pair of classes of $R$-modules

[TABLE]

is a hereditary torsion theory on the category of $R$-modules. In the integral domain case, $w$-modules were called semi-divisorial modules in [3] and (in the ideal case) $F_{\infty}$-ideals in [4], which have been proved to be useful in the study of multiplicative ideal theory and module theory.

In [14], the first named author and Kim generalized projective modules to the hereditary torsion theory ${\rm tor_{\rm GV}}$ setting and introduced the notion of $w$-projective modules. Recall that an $R$-module $M$ is said to be $w$-projective if ${\rm Ext}^{1}_{R}(L(M),N)$ is GV-torsion for any torsionfree $w$-module $N$, where $L(M)=\left(M/{\rm tor_{\rm GV}}(M)\right)_{w}$. It is clear that both ${\rm GV}$-torsion modules and projective modules are $w$-projective. Actually, the notion of $w$-projective modules appeared first in [13] when $R$ is an integral domain. Thus, it is natural to ask if Kaplansky’s theorem on projective modules has a $w$-module theoretic analogue.

To give a $w$-version of Kaplansky’s theorem, we first introduce and study a class of modules closely related to the $w$-projective modules called $w$-split modules (see Section 2). Then we prove, in Section 3, the Kaplansky’s theorem for $w$-projective $w$-modules in terms of $w$-split modules. More precisely, it is shown that every $w$-projective $w$-module can be filtered by countably generated and $w$-split modules (see Theorem 3.5).

Any undefined notions and notations are standard, as in [9, 2, 12].

2. On $w$-split modules

In this section, we introduce and study $w$-split modules, which can be used to prove a $w$-version of Kaplansky’s theorem on projective modules.

Before we give the definition of $w$-split modules, we first fix the following notation. If $M$ is an $R$-module and $s\in R$, then let $\eta^{M}_{s}:M\rightarrow M$ denote the multiplication map $m\mapsto sm$. It is obvious that $\eta^{M}_{1}=\textbf{1}_{M}$ is the identity map on $M$. Moreover, if $M,N$ are $R$-modules, then for any $a\in R$ and any $f\in{\rm Hom}_{R}(M,N)$, we have the multiplication $a\cdot f=f\eta^{M}_{a}=\eta^{N}_{a}f$.

Definition**.**

1. (1)

A short exact sequence of $R$-modules

[TABLE]

is said to be $w$-split if there exist $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and $h_{1},\dots,h_{n}\in{\rm Hom}_{R}(C,B)$ such that $\eta_{d_{k}}^{C}=gh_{k}$ for all $k=1,\dots,n$. 2. (2)

An $R$-module $M$ is said to be $w$-split if there is a $w$-split short exact sequence of $R$-modules

[TABLE]

with $P$ projective.

The first part of the following lemma is exactly [12, Exercise 1.60] (without proof). However, for the purposes of showing the second part of the lemma, we still give a proof for it.

Lemma 2.1**.**

Consider the following commutative diagram with exact rows:

[TABLE]

Then there exists a homomorphism $h:B\rightarrow A^{\prime}$ with $hf=\alpha$ if and only if there is a homomorphism $h^{\prime}:C\rightarrow B^{\prime}$ with $g^{\prime}h^{\prime}=\gamma$. In this case, the equality $\beta=f^{\prime}h+h^{\prime}g$ holds.

Proof.

Assume that there is a homomorphism $h:B\rightarrow A^{\prime}$ such that $hf=\alpha$. Then for any $c\in C$, since $g$ is epic, $c=g(b)$ for some $b\in B$. Define $h^{\prime}:C\rightarrow B^{\prime}$ by

[TABLE]

If $c=g(b)=g(b_{1})$ for some $b_{1}\in B$, then $b-b_{1}\in{\rm ker}(g)={\rm im}(f)$. Therefore, $b-b_{1}=f(a)$ with $a\in A$, and so

[TABLE]

i.e., $\beta(b)-f^{\prime}h(b)=\beta(b_{1})-f^{\prime}h(b_{1})$. Hence, $h^{\prime}$ is a well-defined homomorphism with $g^{\prime}h^{\prime}=\gamma$. Conversely, let $h^{\prime}:C\rightarrow B^{\prime}$ be a homomorphism with $g^{\prime}h^{\prime}=\gamma$. Then for each $b\in B$, we have

[TABLE]

and so $\beta(b)-h^{\prime}g(b)\in{\rm ker}(g^{\prime})={\rm im}(f^{\prime}).$ Since $f^{\prime}$ is monic, there is a unique $a^{\prime}\in A^{\prime}$ with $f^{\prime}(a^{\prime})=\beta(b)-h^{\prime}g(b)$. Now, define $h:B\rightarrow A^{\prime}$ by $h(b)=a^{\prime}$. Then it is easy to check that $h$ is a well-defined homomorphism. For any $a\in A$, write $a^{\prime}=hf(a)$. Then

[TABLE]

Thus, it follows that $hf=\alpha$.

Moreover, the second statement follows immediately by the above proof. ∎

By using the lemma above, it is easy to prove the following proposition.

Proposition 2.2**.**

An exact sequence of $R$-modules $0\rightarrow A\stackrel{{\scriptstyle f}}{{\rightarrow}}B\stackrel{{\scriptstyle g}}{{\rightarrow}}C\rightarrow 0$ is $w$-split if and only if there are $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and $q_{1},\dots,q_{n}\in{\rm Hom}_{R}(B,A)$ such that $\eta_{d_{k}}^{A}=q_{k}f$ for all $k=1,\dots,n$. In this case, for each $k$, the equality $\eta_{d_{k}}^{B}=fq_{k}+h_{k}g$ holds, where $h_{k}$ is as in the definition.

Proof.

Assume that the exact sequence $0\rightarrow A\stackrel{{\scriptstyle f}}{{\rightarrow}}B\stackrel{{\scriptstyle g}}{{\rightarrow}}C\rightarrow 0$ is $w$-split. Then there exist $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and $h_{1},\dots,h_{n}\in{\rm Hom}_{R}(C,B)$ such that $\eta_{d_{k}}^{C}=gh_{k}$ for all $k=1,\dots,n$. For each $k$, let us consider the following commutative diagram with exact rows:

[TABLE]

Thus, by Lemma 2.1, there is a homomorphism $q_{k}:B\rightarrow A$ with $\eta_{d_{k}}^{A}=q_{k}f$ and $\eta_{d_{k}}^{B}=fq_{k}+h_{k}g$.

In a similar way, we can see that the converse is also true. ∎

The following lemma gives a condition under which a GV-torsion module is $w$-split. Also, it will be used to characterize $w$-split modules.

Lemma 2.3**.**

Let $M$ be a ${\rm GV}$-torsion $R$-module. Then $M$ is $w$-split if and only if there exists a $J\in{\rm GV}(R)$ with $JM=0$.

Proof.

If $M$ is a $w$-split module, then there is a $w$-split short exact sequence $0\rightarrow A\rightarrow P\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ with $P$ a projective module. Then we can pick $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and $h_{k}:M\rightarrow P$ as in definition. Since $P$ is ${\rm GV}$-torsionfree, we must have $h_{k}=0$ for all $k=1,\dots,n$. Hence, for each $x\in M$, $d_{k}x=\eta^{M}_{d_{k}}(x)=gh_{k}(x)=0$ for all $k$, and so $JM=0$.

Conversely, suppose that there is a $J\in{\rm GV}(R)$ with $JM=0$. Consider a short exact sequence $0\rightarrow A\rightarrow P\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ of $R$-modules with $P$ projective. Set $J=\langle d_{1},\dots,d_{n}\rangle$ and let $h_{k}:M\rightarrow P$ be the zero maps for all $k=1,\dots,n$. Then for any $x\in M$, we obtain $d_{k}x=0=gh_{k}(x)$, and so $\eta_{d_{k}}^{M}=gh_{k}$ for each $k$. Therefore, $M$ is a $w$-split module. ∎

Proposition 2.4**.**

The following statements are equivalent for an $R$-module $M$.

1. (1)

$M$* is a $w$-split module.* 2. (2)

${\rm Ext}^{1}_{R}(M,N)$* is ${\rm GV}$-torsion for all $R$-modules $N$.* 3. (3)

${\rm Ext}^{i}_{R}(M,N)$* is ${\rm GV}$-torsion for all $R$-modules $N$ and for all integers $i\geq 1$.* 4. (4)

For any $R$-epimorphism $g:B\rightarrow C$, the induced map

[TABLE]

has a ${\rm GV}$-torsion cokernel. 5. (5)

For any $R$-epimorphism $g^{\prime}:F\rightarrow M$ with $F$ projective, the induced map

[TABLE]

has a ${\rm GV}$-torsion cokernel. 6. (6)

For any $R$-epimorphism $g:B\rightarrow C$ and for each homomorphism $\alpha:M\rightarrow C$, there exist $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and homomorphisms $h_{k}:M\rightarrow B$ such that $gh_{k}=d_{k}\cdot\alpha$ where $k=1,\dots,n$. 7. (7)

Every exact sequence $0\rightarrow A\stackrel{{\scriptstyle f}}{{\rightarrow}}B\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ of $R$-modules is $w$-split. 8. (8)

There exist elements $\{x_{i}\}_{i\in I}$ of $M$ and $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ such that for all $k=1,\dots,n$, there are homomorphisms $\{f_{k_{i}}\in M^{*}\}_{i\in I}$ satisfying that for each $x\in M$, almost all $f_{k_{i}}(x)=0$ and $d_{k}x=\sum\limits_{i}f_{k_{i}}(x)x_{i}$.

Proof.

$(1)\Rightarrow(2)$ Suppose that $M$ is a $w$-split module and let $N$ be an $R$-module. Then there exists a $w$-split exact sequence $0\rightarrow L\stackrel{{\scriptstyle f}}{{\rightarrow}}F\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ of $R$-modules with $F$ projective. Since

[TABLE]

is exact, it suffices to prove that ${\rm coker}(f^{*})$ is a ${\rm GV}$-torsion module. But this is equivalent to showing that for any $\alpha\in{\rm Hom}_{R}(L,N)$, there is a $J\in{\rm GV}(R)$ with $J\alpha\subseteq{\rm im}(f^{*})$. Now let $\alpha\in{\rm Hom}_{R}(L,N)$ be arbitrary. Pick $J$ and $q_{k}:F\rightarrow L$ as in Proposition 2.2, and set $\beta_{k}=\alpha q_{k}$, where $k=1,\dots,n$. Then $f^{*}(\beta_{k})=\beta_{k}f=\alpha\eta^{L}_{d_{k}}=d_{k}\cdot\alpha$ for all $k$. Thus, it follows that $J\alpha\subseteq{\rm im}(f^{*})$, as desired.

$(2)\Rightarrow(3)$ It follows from standard homological algebra.

$(3)\Rightarrow(4)$ Suppose that (3) holds and let $g:B\rightarrow C$ be an epimorphism of $R$-modules. Then the sequence

[TABLE]

is exact. Thus, by (3), ${\rm coker}(g_{*})$ is ${\rm GV}$-torsion.

$(4)\Rightarrow(5)$ is trivial.

$(5)\Rightarrow(1)$ This follows easily from the definition of $w$-split modules.

$(4)\Rightarrow(6)$ Assume that (4) holds. Let $g:B\rightarrow C$ be an epimorphism of $R$-modules and $\alpha:M\rightarrow C$ a homomorphism. Then ${\rm coker}(g_{*})$ is a ${\rm GV}$-torsion module, and so there exists a $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ with $J\alpha\subseteq{\rm im}(g_{*})$. Thus, we can find $h_{1},\dots,h_{n}\in{\rm Hom}_{R}(M,B)$ such that for each $k=1,\dots,n$, $d_{k}\cdot\alpha=gh_{k}$, and so (6) holds.

$(6)\Rightarrow(7)$ Apply (6) to the identity map $\textbf{1}_{M}:M\rightarrow M$.

$(7)\Rightarrow(8)$ Let $0\rightarrow A\rightarrow F\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ be an exact sequence of $R$-modules with $F$ free. Then by (7), it is $w$-split, and so there exist $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and $h_{1},\dots,h_{n}\in{\rm Hom}_{R}(M,F)$ such that $\eta_{d_{k}}^{M}=gh_{k}$ for all $k=1,\dots,n$. Now, let $\{e_{i}\}_{i\in I}$ be a basis of $F$ and set $x_{i}=g(e_{i})$ for each $i\in I$. Then for any $x\in M$, $h_{k}(x)=\sum\limits_{i}r_{k_{i}}e_{i}$, where $r_{k_{i}}\in R$ and only finitely many $r_{k_{i}}\neq 0$. Define $f_{k_{i}}:M\rightarrow R$ for all $k=1,\dots,n$ and for all $i\in I$, by $f_{k_{i}}(x)=r_{k_{i}}$. It is clear that all $f_{k_{i}}\in M^{*}$ and, for any $x\in M$, we have $d_{k}x=\sum\limits_{i}f_{k_{i}}(x)x_{i}$. Hence, (8) holds.

$(8)\Rightarrow(1)$ Assume that (8) holds. Define $F$ to be the free $R$-module with basis $\{e_{i}\}_{i\in I}$, and define an $R$-map $g:F\rightarrow M^{\prime}$ by $g:e_{i}\mapsto x_{i}$, where $M^{\prime}$ is the $R$-submodule of $M$ generated by the set $\{x_{i}\}_{i\in I}$. Then we obtain an exact sequence $0\rightarrow A\rightarrow F\stackrel{{\scriptstyle g}}{{\rightarrow}}M^{\prime}\rightarrow 0$ with $A={\rm ker}(g)$. For each $k=1,\dots,n$, define $h_{k}:M^{\prime}\rightarrow F$ by $h_{k}(y)=\sum\limits_{i}f_{k_{i}}(y)e_{i}$, where $y\in M^{\prime}$. Since the sum is finite, $h_{k}$ is a well-defined homomorphism. Moreover, for any $y\in M^{\prime}$,

[TABLE]

i.e., $\eta_{d_{k}}^{M^{\prime}}=gh_{k}$. Therefore, $M^{\prime}$ is a $w$-split module.

Finally, notice that $J(M/M^{\prime})=0$ and $M/M^{\prime}$ is a GV-torsion $R$-module. Then $M/M^{\prime}$ is $w$-split by Lemma 2.3. Thus, the equivalence of (1) and (2) implies that $M$ is also $w$-split. ∎

As a corollary of Proposition 2.4, we have:

Corollary 2.5**.**

The following statements hold.

1. (1)

Every $w$-split module is $w$-projective. 2. (2)

Let $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be an exact sequence of $R$-modules with $C$ $w$-split. Then $A$ is $w$-split if and only if so is $B$.

Next, we will give an example of a $w$-projective module, which is not $w$-split.

Example 2.6*.*

Let $R$ be a two dimensional regular local ring with the maximal ideal $\mathfrak{m}$ and set

[TABLE]

Then $M$ is a ${\rm GV}$-torsion module, and so it is $w$-projective. Now, we say that $M$ is not $w$-split. If not, then there is a $J_{0}\in{\rm GV}(R)$ with $J_{0}M=0$ by Lemma 2.3. This means that $J_{0}$ is contained in all $J\in{\rm GV}(R)$. We claim next that $\mathfrak{m}\in{\rm GV}(R)$. Indeed, since $R$ is a two dimensional regular local ring, $\mathfrak{m}$ is generated by an $R$-sequence of length two, and so

[TABLE]

by [6, p. 102, Exercise 1], where $\mathfrak{m}^{-1}=\{x\in Q~{}|~{}\mathfrak{m}x\subseteq R\}$ and $Q$ is the quotient field of $R$. Therefore, $\mathfrak{m}\in{\rm GV}(R)$, and so is $\mathfrak{m}^{n}$ for any integer $n\geq 1$. Thus, it follows that

[TABLE]

whence $J_{0}=0$. However, this means that $R$ as a module over itself is both GV-torsion and GV-torsionfree, hence $R=0$, which is a contradiction.

We close this section with a short discussion of when a $w$-projective module is $w$-split.

Proposition 2.7**.**

Every $w$-projective $w$-module is $w$-split.

Proof.

Let $M$ be a $w$-projective $w$-module over $R$ and let $g:P\rightarrow M$ be an $R$-epimorphism with $P$ projective. It follows that $0\rightarrow K\rightarrow P\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ is exact, where $K={\rm ker}(g)$. Then we have the following exact sequence

[TABLE]

Since $M$ is ${\rm GV}$-torsionfree, $K$ is a torsionfree $w$-module, and so ${\rm Ext}^{1}_{R}(M,K)$ is ${\rm GV}$-torsion by the $w$-projectivity of $M$. Thus, ${\rm coker}(g_{*})$ is also ${\rm GV}$-torsion, whence $M$ is a $w$-split module by Proposition 2.4. ∎

Proposition 2.8**.**

Let $M$ be a ${\rm GV}$-torsionfree $w$-projective $R$-module. Then $M$ is $w$-split if and only if there is a $J\in{\rm GV}(R)$ with $JM_{w}\subseteq M$.

Proof.

Write $T=M_{w}/M$. Then $T$ is a ${\rm GV}$-torsion module and the sequence $0\rightarrow M\stackrel{{\scriptstyle\mu}}{{\rightarrow}}M_{w}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}T\rightarrow 0$ is exact, where $\mu$ is the inclusion map and $\pi$ is the natural map. Also, note that $L(M)=M_{w}=L(M_{w})$. Thus it follows, from the definition of $w$-projective modules and the $w$-projectivity of $M$, that $M_{w}$ is also $w$-projective. Hence, it follows from Proposition 2.7 that $M_{w}$ is $w$-split.

If there is a $J\in{\rm GV}(R)$ with $JM_{w}\subseteq M$, then $JT=0$, and so $T$ is $w$-split by Lemma 2.3. Therefore, Corollary 2.5(2) says that $M$ is $w$-split.

To prove the converse, it suffices by Lemma 2.3 to show that $T$ is a $w$-split module. This in turn is equivalent to prove that for each $R$-module $N$, ${\rm Ext}^{1}_{R}(T,N)$ is ${\rm GV}$-torsion. Let $N$ be an arbitrary $R$-module. Thus, consider the following exact sequence

[TABLE]

Then since $M_{w}$ is $w$-split, ${\rm Ext}^{1}_{R}(M_{w},N)$ is ${\rm GV}$-torsion. Hence, to complete the proof, we need only to show that ${\rm coker}(\mu^{*})$ is ${\rm GV}$-torsion as well. As $M$ is a $w$-split module, there is a $w$-split short exact sequence $0\rightarrow K\rightarrow P\stackrel{{\scriptstyle g}}{{\rightarrow}}M\rightarrow 0$ with $P$ projective, hence there exists $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$ and homomorphisms $h_{1},\dots,h_{n}:M\rightarrow P$ such that $\eta_{d_{k}}^{M}=gh_{k}$, for all $k=1,\dots,n$. Now, let $\alpha\in{\rm Hom}_{R}(M,N)$ and consider the following commutative diagram with exact rows

[TABLE]

where $P$ and $Q$ are projective modules. Since $P$ is a $w$-module, each $h_{k}$ can be extended to a homomorphism $h^{\prime}_{k}:M_{w}\rightarrow P$. For any $k$, set $f_{k}=g^{\prime}\beta h^{\prime}_{k}$. Then it is easy to check that $\mu^{*}(f_{k})=d_{k}\cdot\alpha$ for all $k$. Hence, it follows that $J\alpha\subseteq{\rm im}(\mu^{*})$, i.e., ${\rm coker}(\mu^{*})$ is ${\rm GV}$-torsion. ∎

3. Kaplansky’s theorem for $w$-projective $w$-modules

To begin with, we fix for all the section the following notation. Let

[TABLE]

be a $w$-split exact sequence of $R$-modules and $F_{1}$ a submodule of $F$. Then pick $J\in{\rm GV}(R)$ and $h_{1},\dots,h_{n}\in{\rm Hom}_{R}(M,F)$ as in the definition in Section 2 and write

[TABLE]

Thus, the sequence

[TABLE]

is also exact. Moreover, if for each $k=1,\dots,n$, $h_{k}(M_{1})\subseteq F_{1}$, then $(\xi_{1})$ is also $w$-split. In this case ($h_{k}(M_{1})\subseteq F_{1}$ for all $k=1,\dots,n$), we call $(\xi_{1})$ a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $F_{1}$.

Let $F$ be as in $(\xi)$ with a direct sum decomposition $F=\bigoplus\limits_{i\in I}F_{i}$ of projective submodules, where $I$ is an index set. For each subset $H$ of $I$, if $H=\emptyset$, then write

[TABLE]

otherwise, write

[TABLE]

[TABLE]

It is obvious that $F(I)=F$, $M(I)=M$ and $P(I)=P$, and that if $H_{1},H_{2}$ are subsets of $I$ with $H_{1}\subseteq H_{2}$, then $F(H_{1})$ is a direct summand of $F(H_{2})$, $P(H_{1})\subseteq P(H_{2})$, and $M(H_{1})\subseteq M(H_{2})$. Now, $M(H)$ is said to be a $w$-split module induced by $(\xi)$ with respect to the subset $H$ if the sequence

[TABLE]

is a $w$-split exact sequence induced by $(\xi)$ with respect to the projective submodule $F(H)$.

Lemma 3.1**.**

Let $F$ be the module as in the previous paragraph. Suppose that

[TABLE]

is a set, totally ordered by inclusion, satisfying the sequence

[TABLE]

is a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $F(H_{s})$ for any $H_{s}\in\mathcal{S}_{1}$. Set $H=\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}H_{s}$. Then the following statements hold.

1. (1)

$\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}F(H_{s})=F(H)$* and hence it is a projective module. Moreover, $\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}P(H_{s})=P(H)$ and $\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}M(H_{s})=M(H)$.* 2. (2)

The sequence

[TABLE]

is a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}F(H_{s})$. Hence $\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}M(H_{s})$ is a $w$-split module induced by $(\xi)$ with respect to the subset $H$.

Proof.

(1) For each $y\in F(H)$, write $y=\sum\limits_{j\in H}y_{j}$, where $y_{j}\neq 0$ for only a finite number of indices $j$. Since $\mathcal{S}_{1}$ is totally ordered, we can choose some $H_{s_{0}}\in\mathcal{S}_{1}$ such that $y=\sum\limits_{i\in H_{s_{0}}}y_{i}\in F(H_{s_{0}})$, and so $F(H)\subseteq\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}F(H_{s})$. The other inclusion is clear.

It is obvious that $\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}P(H_{s})\subseteq P(H)$. For the other inclusion, let $x\in P(H)$. Then we have

[TABLE]

Since $F(H)=\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}F(H_{s})$, there is some $H_{s_{0}}\in\mathcal{S}_{1}$ such that $f(x)\in F(H_{s_{0}})$, and so $f(x)\in{\rm ker}(g)\bigcap F(H_{s_{0}})={\rm ker}(g_{H_{s_{0}}})$. Thus, $x\in f^{-1}({\rm ker}(g_{H_{s_{0}}}))=P(H_{s_{0}})$, whence $P(H)\subseteq\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}P(H_{s})$.

Let $x\in M(H)$. Then $x=g(y)$ for some $y\in F(H)$. There is some $H_{s_{0}}\in\mathcal{S}_{1}$ such that $y\in F(H_{s_{0}})$. Therefore, $x\in g\left(F(H_{s_{0}})\right)=M(H_{s_{0}})$, whence $M(H)\subseteq\mathop{\bigcup}\limits_{H_{s}\in\mathcal{S}_{1}}M(H_{s})$. The other inclusion is obvious.

(2) Pick $h_{1},\dots,h_{n}\in{\rm Hom}_{R}(M,F)$ as in the definition of $w$-split exact sequences. Then we need only show that

[TABLE]

for all $k=1,\dots,n$. Now let $x\in\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}M(H_{s})$ be arbitrary. Then $x\in M(H_{s_{0}})$ for some $H_{s_{0}}\in\mathcal{S}_{1}$. Since $\left(\xi_{H_{s_{0}}}\right)$ is a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $F(H_{s_{0}})$, we have $h_{k}\left(M(H_{s_{0}})\right)\subseteq F(H_{s_{0}})$ for any $k$, and so $h_{k}(x)\in F(H_{s_{0}})\subseteq\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}F(H_{s})$, as desired.

The second assertion of (2) is clear. ∎

Recall that an ideal $I$ of $R$ is said to be a $w$-ideal if it is a $w$-module as an $R$-module, i.e., $I=I_{w}$. Let $w\mbox{-}{\rm Max}(R)$ denote the set of $w$-ideals of $R$ maximal among proper integral $w$-ideals of $R$ and we call $\mathfrak{m}\in w\mbox{-}{\rm Max}(R)$ a maximal $w$-ideal of $R$. Then every proper $w$-ideal is contained in a maximal $w$-ideal and every maximal $w$-ideal is a prime ideal.

Let $M$ be an $R$-module and $S$ a multiplicatively closed subset of $R$. Then we denote as usual by $M_{S}$ the localization of $M$ at $S$. In particular, if $\mathfrak{p}$ is a prime ideal of $R$ and $S=R\backslash\mathfrak{p}$, then $M_{S}$ is denoted by $M_{\mathfrak{p}}$. Let $f:M\rightarrow N$ be a homomorphism of $R$-modules. For $s\in S$ and $x\in M$, define

[TABLE]

Then $f_{S}:M_{S}\rightarrow N_{S}$ is a well-defined $R_{S}$-homomorphism.

Recall from [12] that an $R$-homomorphism $f:M\rightarrow N$ is called a $w$-isomorphism if $f_{\mathfrak{m}}:M_{\mathfrak{m}}\rightarrow N_{\mathfrak{m}}$ is an isomorphism over $R_{\mathfrak{m}}$ for any $\mathfrak{m}\in w\mbox{-}{\rm Max}(R)$. Since an $R$-module $M$ is ${\rm GV}$-torsion if and only if $M_{\mathfrak{m}}=0$ for any $\mathfrak{m}\in w\mbox{-}{\rm Max}(R)$ (see [12, Theorem 6.2.15]), it is easy to see that a homomorphism $f:M\rightarrow N$ is a $w$-isomorphism if and only if both ${\rm ker}(f)$ and ${\rm coker}(f)$ are ${\rm GV}$-torsion.

Now, we call an $R$-module $M$ a $w$-countably generated module if there is a $w$-isomorphism $f:M_{0}\rightarrow M$ with $M_{0}$ a countably generated $R$-module. It is easily seen that $M$ is $w$-countably generated if and only if there exists a submodule $N$ of $M$ such that for any $\mathfrak{m}\in w\mbox{-}{\rm Max}(R)$, $N_{\mathfrak{m}}=M_{\mathfrak{m}}$.

Lemma 3.2**.**

Let $F$ be as in $(\xi)$. Suppose that $F$ is a $w$-module over $R$ with a direct sum decomposition $F=\bigoplus\limits_{i\in I}F_{i}$ of countably generated submodules. For each subset $H$ of $I$, let $F(H)$, $P(H)$ and $M(H)$ be as before. If $H$ is a proper subset of $I$ satisfying $(\xi_{H})$ is a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $F(H)$, then the following statements hold.

1. (1)

There is a subset $H_{1}$ of $I$ properly containing $H$ such that

[TABLE]

is a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $F(H_{1})$. 2. (2)

$C:=M(H_{1})/M(H)$* is a countably generated module.* 3. (3)

If $M$ is ${\rm GV}$-torsionfree, then $C$ is $w$-isomorphic to $D:=M(H_{1})_{w}/M(H)_{w}$. In this case, $D$ is a $w$-countably generated module. 4. (4)

If $M$ is ${\rm GV}$-torsionfree and if each $F_{i}$ is projective, then $M(H)$ and $M(H_{1})$ are $w$-split modules induced by $(\xi)$ with respect to the subsets $H$ and $H_{1}$, respectively, and $C$ is $w$-split. In this case, $D$ is a $w$-countably generated and $w$-projective module.

Proof.

(1) Since $(\xi)$ is a $w$-split exact sequence, we can pick $J=\langle d_{1},\dots,d_{n}\rangle\in{\rm GV}(R)$, $h_{k}:M\rightarrow F$ and $q_{k}:F\rightarrow P$ as in Proposition 2.2, where $k=1,\dots,n$. Then for any $j\in I$, both $fq_{k}(F_{j})$ and $h_{k}g(F_{j})$ are countably generated modules, and for each $x\in F_{j}$, we have $d_{k}x=fq_{k}(x)+h_{k}g(x)$. Therefore, $d_{k}F_{j}\subseteq fq_{k}(F_{j})+h_{k}g(F_{j})$ for all $k$.

Choose an $i_{0}\in I\backslash H$. Then there exists a countable subset $I_{1}$ of $I$ such that

[TABLE]

for all $k$. Note that $\bigoplus\limits_{i\in I_{1}}F_{i}$ is countably generated as each $F_{i}$ is countably generated and $I_{1}$ is a countable set. Thus, we can find another countable subset $I_{2}$ of $I$ containing $I_{1}$ with

[TABLE]

for all $k$. Continuing, we obtain countable subsets $I_{0}=\{i_{0}\},I_{1},I_{2},\dots,I_{s},\dots$ satisfying

[TABLE]

for all $k$. Hence,

[TABLE]

But since $J_{w}=R$ (cf. [15, Proposition 3.5]) and both $\bigoplus\limits_{j\in I_{s}}F_{j}$ and $\bigoplus\limits_{i\in I_{s+1}}F_{i}$ are $w$-modules (cf. [15, Proposition 2.3]), it follows easily from [12, Theorem 6.2.2] that $\bigoplus\limits_{j\in I_{s}}F_{j}\subseteq\bigoplus\limits_{i\in I_{s+1}}F_{i}$.

Set $L=\bigcup\limits_{s=0}^{\infty}I_{s}$. Then it is a countable set, and so $L_{1}:=L\backslash H$ is countable too. Write $H_{1}=H\bigcup L$. Then $H_{1}=H\bigcup L_{1}$. Thus $V:=\bigoplus\limits_{i\in L_{1}}F_{i}$ is countably generated and $F(H_{1})=F(H)\bigoplus V$. For each $j\in H_{1}$, if $j\in H$, then since $(\xi_{H})$ is a $w$-split exact sequence induced by $(\xi)$ with respect to the submodule $F(H)$, we obtain

[TABLE]

Otherwise, $j\in L_{1}$, and so $j\in I_{s}$ for some $s$. Hence, by $({\dagger})$, we also have $h_{k}g(F_{j})\subseteq F(H_{1})$. So it follows that $h_{k}(M(H_{1}))\subseteq F(H_{1})$, whence (1) holds.

(2) Consider the following commutative diagram with exact rows and columns

[TABLE]

where $V$ is as in the proof of (1). Then since $V$ is countably generated, so is $C$.

(3) Consider the following commutative diagram having exact rows.

[TABLE]

Then the Snake Lemma implies that the sequence

[TABLE]

is exact. Therefore, both ${\rm ker}(h)$ and ${\rm coker}(h)$ are ${\rm GV}$-torsion, i.e., $h$ is a $w$-isomorphism. Hence, by (2), $D$ is $w$-countably generated.

(4) If each $F_{i}$ is a projective module, then it is clear that $M(H)$ and $M(H_{1})$ are $w$-split modules induced by $(\xi)$ with respect to the subsets $H$ and $H_{1}$, respectively. Let $J=\langle d_{1},\dots,d_{n}\rangle$ and $h_{k}$ ($k=1,\dots,n$) as in the proof of (1). Then to show that $C$ is $w$-split, let us consider, for each $k$, the following diagram having exact rows

[TABLE]

where $h^{(k)}=h_{k}|_{M(H)}$, $h^{(k)}_{1}=h_{k}|_{M(H_{1})}$, and $V$ is as in the proof of (1). Clearly, the left square commutes, and so there is a homomorphism $\alpha_{k}:C\rightarrow V$ such that the right square commutes as well. Let $g^{\prime}$ be as in the proof of (2). Then it is not difficult to see that $g^{\prime}\alpha_{k}=\eta^{C}_{d_{k}}$ for all $k$. Thus, $0\rightarrow A\rightarrow V\stackrel{{\scriptstyle g^{\prime}}}{{\longrightarrow}}C\rightarrow 0$ is a $w$-split exact sequence with $V$ projective, and consequently $C$ is $w$-split. ∎

Let $\alpha$ be an ordinal and $\mathcal{A}=(A_{\lambda}~{}|~{}\lambda\leq\alpha)$ a sequence of modules. Then $\mathcal{A}$ is called a continuous chain of modules (see [2]) if $A_{0}=0$, $A_{\lambda}\subseteq A_{\lambda+1}$ for all $\lambda<\alpha$ and $A_{\lambda}=\bigcup\limits_{\mu<\lambda}A_{\mu}$ for all limit ordinals $\lambda\leq\alpha$.

Let $M$ be a $w$-split $R$-module. Then there exists a $w$-split exact sequence $(\xi)$ with $F$ projective. Write $F=\bigoplus\limits_{i\in I}F_{i}$, where each $F_{i}$ is a countably generated projective module. Now, a submodule $N$ of $M$ is said to be filtered by countably generated $w$-split modules if, for some ordinal $\alpha$, there exist a subset $H$ of $I$ and a sequence

[TABLE]

of subsets of $H$ such that $H_{0}=\emptyset$, $H_{\alpha}=H$ and

1. (i)

the sequence of modules

[TABLE]

is a continuous chain of $N$ with $N_{\alpha}=N$; 2. (ii)

each $N_{\lambda}$ is a $w$-split module induced by $(\xi)$ with respect to the subset $H_{\lambda}$; 3. (iii)

for each $\lambda<\alpha$, $N_{\lambda+1}/N_{\lambda}$ is a countably generated $w$-split module.

The chain $\mathcal{N}$ is called a countably generated $w$-split filtration of $N$.

In this case, if $B$ is also a submodule of $M$ and if, for some ordinal $\beta\leq\alpha$, $B$ is filtered by countably generated $w$-split modules having a continuous chain

[TABLE]

of submodules such that $B_{\mu}=N_{\mu}$ for each $\mu\leq\beta$, then we call $N$ a filtered extension of $B$ by countably generated $w$-split modules.

Lemma 3.3**.**

Let, as in the $(\xi)$ above, $M$ be a $w$-split module and $F$ a projective module with a direct sum decomposition $F=\bigoplus\limits_{i\in I}F_{i}$ of countably generated projective submodules. Suppose that $\{H_{j}\}_{j\in\Gamma}$ is a totally ordered family of subsets of $I$ satisfying that for each $j\in\Gamma$, $A_{j}:=M(H_{j})$ can be filtered by countably generated $w$-split modules, and that for $j,k\in\Gamma$, if $H_{j}\subseteq H_{k}$, then $A_{k}$ is a filtered extension of $A_{j}$ by countably generated $w$-split modules. Then $N:=\bigcup\limits_{j\in\Gamma}A_{j}$ can also be filtered by countably generated $w$-split modules and for each $j\in\Gamma$, it is a filtered extension of $A_{j}$.

Proof.

Notice that, if $N=A_{j}$ for some $j\in\Gamma$, then we have nothing to prove. So we assume that for any $j\in\Gamma$, $N\neq A_{j}$. For all $j\in\Gamma$, since $A_{j}$ is filtered by countably generated $w$-split modules, there are an ordinal $\alpha_{j}$ and $H_{j}$ a subset of $I$ such that

[TABLE]

is a countably generated $w$-split filtration of $A_{j}$, where $\left(H^{(j)}_{\lambda_{j}}~{}|~{}\lambda_{j}\leq\alpha_{j}\right)$ is a sequence of subsets of $H_{j}$ with $H^{(j)}_{0}=\emptyset$ and $H^{(j)}_{\alpha_{j}}=H_{j}$.

Set $\alpha=\mathop{\bigcup}\limits_{j\in\Gamma}\alpha_{j}$. Then we claim that $\alpha\notin\{\alpha_{j}\}_{j\in\Gamma}$, and so it is a limit ordinal. Otherwise, $\alpha=\alpha_{j}$ for some $j\in\Gamma$. Since $N\neq A_{j}$, there exists a $k\in\Gamma$ with $A_{k}\nsubseteq A_{j}$, whence $A_{j}\subset A_{k}$. Hence, it follows, from the definition of filtered extensions above, that

[TABLE]

which is a contradiction.

Set $H=\mathop{\bigcup}\limits_{j\in\Gamma}H_{j}$. Then we construct a sequence

[TABLE]

of subsets of $H$ as follows.

1. (1)

Write $H_{0}=\emptyset$ and $H_{\alpha}=H$. 2. (2)

Let $0<\lambda<\alpha$. Then we must have some $j\in\Gamma$ with $\lambda<\alpha_{j}$. Define $H_{\lambda}=H^{(j)}_{\lambda}$.

Next, we prove that the sequence of submodules

[TABLE]

is a countably generated $w$-split filtration of $N$.

1. (i)

The sequence $\mathcal{N}$ is a a continuous chain of $N$ with $N_{\alpha}=N$.

1. (a)

$N_{0}=M(H_{0})=0$ is clear. 2. (b)

$N_{\alpha}=N$.

By Lemma 3.1, we have

[TABLE] 3. (c)

$N_{\lambda}\subseteq N_{\lambda+1}$ for all $\lambda<\alpha$.

If $\lambda<\alpha$, then $\lambda+1<\alpha$ as $\alpha$ is a limit ordinal. Consequently, $\lambda<\lambda+1<\alpha_{j}$ for some $j\in\Gamma$, and hence

[TABLE] 4. (d)

$N_{\lambda}=\bigcup\limits_{\beta<\lambda}N_{\beta}$ for all limit ordinals $\lambda\leq\alpha$.

Note that we need only consider the case $\lambda=\alpha$. For each $j\in\Gamma$, we have some $k$ with $\alpha_{j}<\alpha_{k}$, and so $A_{j}\subset A_{k}$. It follows that

[TABLE]

This means that $\bigcup\limits_{\beta<\alpha}N_{\beta}$ contains all $A_{j}$’s, whence $\bigcup\limits_{\beta<\alpha}N_{\beta}=N=N_{\alpha}$. 2. (ii)

Each $N_{\lambda}$ is a $w$-split module induced by $(\xi)$ with respect to the subset $H_{\lambda}$.

Clearly, this is true for any $\lambda<\alpha$. For the case $\lambda=\alpha$, it is proved in Lemma 3.1(3). 3. (iii)

For each $\lambda<\alpha$, $N_{\lambda+1}/N_{\lambda}$ is a countably generated $w$-split module.

The proof is similar to that of (c) of (i).

Thus, we see that $N$ is also filtered by countably generated $w$-split modules. While the second assertion follows immediately from the constructions of $\mathcal{H}$ and $\mathcal{N}$. ∎

Remark 3.4*.*

For the construction of $\mathcal{H}$ in the proof of Lemma 3.3, if $0<\lambda<\alpha$ and if there are $j_{1},j_{2}\in\Gamma$ such that $\lambda<\alpha_{j_{1}}$ and $\lambda<\alpha_{j_{2}}$, then either $H_{j_{1}}\subseteq H_{j_{2}}$ or $H_{j_{2}}\subseteq H_{j_{1}}$ as $\{H_{j}\}_{j\in\Gamma}$ is totally ordered, and so either $A_{j_{1}}\subseteq A_{j_{2}}$ or $A_{j_{2}}\subseteq A_{j_{1}}$. However, in both cases, from the definition of filtered extensions, it is easy to see that

[TABLE]

Thus, $H_{\lambda}$ can be defined to be any of $H^{(j_{1})}_{\lambda}$ and $H^{(j_{2})}_{\lambda}$.

Similarly, if $M$ is a $w$-projective $w$-module over $R$ and if for some ordinal $\alpha$, there is a continuous chain

[TABLE]

of $w$-projective $w$-submodules such that $M^{\prime}_{\alpha}=M$ and $M^{\prime}_{\lambda+1}/M^{\prime}_{\lambda}$ is a $w$-countably generated $w$-projective module for each $\lambda<\alpha$, then $M$ is said to be filtered by $w$-countably generated $w$-projective modules.

Now, we can prove the Kaplansky’s theorem for $w$-projective $w$-modules.

Theorem 3.5**.**

Let $M$ be a $w$-projective $w$-module. Then

1. (1)

$M$* can be filtered by countably generated $w$-split modules.* 2. (2)

$M$* can be filtered by $w$-countably generated $w$-projective modules.*

Proof.

(1) Since $M$ is a $w$-projective $w$-module, it is $w$-split by Proposition 2.7. Let as before $(\xi)$ be the $w$-split exact sequence with $F=\bigoplus\limits_{i\in I}F_{i}$ a projective module, where each $F_{i}$ is a countably generated projective module. With the same notation as in Lemma 3.1, let $\mathcal{S}$ be a collection of subsets $H$ of $I$ satisfying:

1. (a)

$M(H)$ is a $w$-split module induced by $(\xi)$ with respect to the subset $H$; 2. (b)

$M(H)$ can be filtered by countably generated $w$-split modules.

Clearly, $\mathcal{S}$ is non-empty as it contains $\emptyset$. Define a partial order $\leqslant$ on $\mathcal{S}$ by $H_{1}\leqslant H_{2}\Leftrightarrow H_{1}\subseteq H_{2}$ and $M(H_{2})$ is a filtered extension of $M(H_{1})$ by countably generated $w$-split modules. Let \mathcal{S}_{1}$$=\{H_{s}\} be a totally ordered subset of $\mathcal{S}$ and $H=\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}H_{s}$. Then Lemma 3.1 says that $M(H)=\bigcup\limits_{H_{s}\in\mathcal{S}_{1}}M(H_{s})$ is a $w$-split module induced by $(\xi)$ with respect to the subset $H$.

Now, we claim that the second condition of Lemma 3.3 is also satisfied. Indeed, as $\mathcal{S}_{1}$ is totally ordered by $\leqslant$, it is also totally ordered by inclusion. Moreover, by the choice of $\mathcal{S}$, for each $H_{s}\in\mathcal{S}_{1}\subseteq\mathcal{S}$, $M(H_{s})$ can be filtered by countably generated $w$-split modules. Also, if $H_{s_{1}},H_{s_{2}}\in\mathcal{S}_{1}$ with $H_{s_{1}}\subseteq H_{s_{2}}$, then either $H_{s_{1}}\leqslant H_{s_{2}}$ or $H_{s_{2}}\leqslant H_{s_{1}}$. In the first case, the definition of $\leqslant$ implies that $M(H_{s_{2}})$ is a filtered extension of $M(H_{s_{1}})$ by countably generated $w$-split modules. In the second case, $H_{s_{2}}\leqslant H_{s_{1}}$, we have $H_{s_{2}}\subseteq H_{s_{1}}$, whence $H_{s_{1}}=H_{s_{2}}$ and $M(H_{s_{1}})=M(H_{s_{2}})$. But it is obvious that $M(H_{s_{2}})$ is a filtered extension of itself ($M(H_{s_{1}})$) by countably generated $w$-split modules.

Thus, it follows from Lemma 3.3 that $M(H)$ can be filtered by countably generated $w$-split modules and that for each $H_{s}\in\mathcal{S}_{1}$, $M(H)$ is a filtered extension of $M(H_{s})$ by countably generated $w$-split modules. Thus, $H\in\mathcal{S}$ and it is a upper bound of $\mathcal{S}_{1}$. By Zorn’s Lemma, $\mathcal{S}$ has a maximal element, say, $H$.

If $H\neq I$, then by Lemma 3.2, there is a subset $H_{1}$ of $I$ properly containing $H$ such that $M(H_{1})$ is a $w$-split module induced by $(\xi)$ with respect to the subset $H_{1}$ and that $C=M(H_{1})/M(H)$ is a countably generated $w$-split module. Therefore, if $\left(M_{\lambda}(H)~{}|~{}\lambda\leq\alpha\right)$ is a countably generated $w$-split filtration of $M(H)$, then by setting $M_{\alpha+1}(H_{1})=M(H_{1})$ and $M_{\lambda}(H_{1})=M_{\lambda}(H)$ for $\lambda\leq\alpha$, we see that $\left(M_{\lambda}(H_{1})~{}|~{}\lambda\leq\alpha+1\right)$ is a countably generated $w$-split filtration of $M(H_{1})$. Hence, $H_{1}\in\mathcal{S}$ and $H\leqslant H_{1}$, which contradicts the maximality of $H$. Therefore, $H=I$ and $M(H)=M$, whence $M$ can be filtered by countably generated $w$-split modules.

(2) By (1), $M$ can be filtered by countably generated $w$-split modules with a continuous chain $(M_{\lambda}~{}|~{}\lambda\leq\alpha)$. For each ordinal $\lambda$, set $M^{\prime}_{\lambda}=(M_{\lambda})_{w}$. Then by Corollary 2.5(1), $M_{\lambda}$ is $w$-projective, and hence $M^{\prime}_{\lambda}$ is a $w$-projective $w$-submodule of $M$. By the similar proof of Lemma 3.2(3), we have that $M_{\lambda+1}/M_{\lambda}$ is $w$-isomorphic to $M^{\prime}_{\lambda+1}/M^{\prime}_{\lambda}$, and so $M^{\prime}_{\lambda+1}/M^{\prime}_{\lambda}$ is $w$-countably generated $w$-projective module. Thus, it follows that $M$ can be filtered by $w$-countably generated $w$-projective modules. ∎

Acknowledgments

The authors would like to thank the referee for the helpful comments and suggestions which substantially improved the paper. This work was partially supported by NSFC (Nos. 11671283 and 11701398) and the Scientific Research Fund of Sichuan Provincial Education Department (No. 17ZB0362).