Fact 2.6 [Reference Guil Asensio, Kalebogaz and Srivastava16, 2.5].

Let $M, N$ be pure-injective modules. If there are $f: M \to N$ a pure embedding and $g: N \to M$ a pure embedding, then M and N are isomorphic.

Fact 2.7.

• • If N is $\Sigma $ -pure-injective, then N is pure-injective.

• • If N is $\Sigma $ -pure-injective and $M \leq _{p} N$ , then M is $\Sigma $ -pure-injective.

• • If N is $\Sigma $ -pure-injective and M is elementary equivalent to N, then M is $\Sigma $ -pure-injective.

• • [Reference Prest29, 3.2] If N is $\Sigma $ -pure-injective, then $(\text {Mod}(Th(N)), \leq _p)$ is $\lambda $ -stable for every $\lambda \geq |Th(N)|$ . For the models of the theory of N, being a pure submodule is the same as being an elementary submodule by $pp$ -quantifier elimination.

Proof. Joint embedding and no maximal models follow directly from closure under direct sums. So we show the amalgamation property.

Let $M \leq _p N_1, N_2$ be models of K. By minimality of the pure-injective envelope we obtain that $PE(M) \leq _p PE(N_1), PE(N_2)$ and observe that all of these models are in K by closure under pure-injective envelopes.

Let $L := PE(N_1) \oplus PE(N_2)$ which is in K by closure under direct sums. Now, as $PE(M)$ is pure-injective, there are $N_1'$ and $N_2'$ such that $PE(N_i)= PE(M) \oplus N_i'$ for $i \in \{ 1, 2\}$ . Hence, $L = (PE(M) \oplus N_1') \oplus (PE(M) \oplus N_2')$ . Define $f: N_1 \to L$ by $f(m + n_1)=(m, n_1, m, 0)$ for $m \in PE(M)$ and $n_1 \in N_1'$ and $g: N_2 \to L$ by $g(m + n_2)=(m, 0, m, n_{2})$ for $m \in PE(M)$ and $n_2 \in N_2'$ . One can show that $f, g$ are pure embeddings such that $f\mathord {\upharpoonright }{M}= g\mathord {\upharpoonright }{M}$ .⊣

Proof. Let $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}=\lambda $ and $M \in \mathbf {K}_\lambda $ . Let $\{ p_i : i < \kappa \}$ be an enumeration without repetitions of $\mathbf {gS}(M)$ . For every $i < \kappa $ fix a pair $(a_i, M_i)$ such that $p_i=\mathbf {gtp}(a_i/M; M_i)$ . Let $\Delta \subseteq \kappa $ such that $| \Delta | \leq 2^{|R| + \aleph _0}$ and for every $i < \kappa $ there is a $j \in \Delta $ such that $M_i$ is elementarily equivalent to $M_j$ . This is possible because there are at most $2^{|R| + \aleph _0}$ complete theories over R. For every $j \in \Delta$ , let $N_j$ be such that $M_j \leq_p N_j$ and $N_j$ is universal over $M_j$ in $(R\text{-Mod}, \leq_p)$ .

Let $\Phi : \kappa \to \bigcup _{j \in \Delta } S_{pp}^{Th(N_j)}(M)$ be such that $\Phi (i)= pp(a_i/M, M_i)$ . By the choice of $\Delta $ and the hypothesis that Galois-types in $\mathbf {K}$ are $pp$ -syntactic we have that $\Phi $ is a well-defined injective function, so $\kappa \leq |\bigcup _{j \in \Delta } S_{pp}^{Th(N_j)}(M)|$ . Observe that for every $j \in \Delta $ we have that $|S_{pp}^{Th(N_j)}(M)|=|S^{Th(N_j)}(M)|$ by $pp$ -quantifier elimination (see [Reference Prest29, Section 2.4]). Hence $|\bigcup _{j \in \Delta } S_{pp}^{Th(N_j)}(M)|= |\bigcup _{j \in \Delta } S^{Th(N_j)}(M)|\leq \sum _{j \in \Delta } | S^{Th(N_j)}(M)|$ . Since every complete first-order theory of modules is $\lambda $ -stable if $\lambda ^{|R| + \aleph _0}=\lambda $ by [Reference Prest29, 3.1] and $|\Delta | \leq 2^{|R| + \aleph _0}$ , we have that $\sum _{j \in \Delta } | S^{Th(N_j)}(M)| \leq \lambda $ . Hence, $\kappa \leq \lambda $ . Therefore, $\mathbf {K}$ is $\lambda $ -stable.⊣

Proof. The forward direction is trivial so we show the backward direction. As K has the amalgamation property we may assume that $N_1=N_2$ and since K is closed under pure-injective envelopes we may assume that $N_1 = N_2$ is pure-injective. Let $N = N_1 =N_2$ . Then by [Reference Ziegler48, 3.6] there is

$$ \begin{align*} h: H^{N}(M \cup \{\bar{b}_1 \}) \cong_{M} H^{N}(M \cup \{ \bar{b}_2 \}) \end{align*} $$

with $h(\bar {b}_1)= \bar {b}_2$ where $H^{N}(M \cup \{\bar {b}_i\})$ denotes the pure-injective envelope of $M \cup \{\bar {b}_i\}$ inside N for every $i \in \{ 1, 2 \}$ .

As it might be the case that $H^{N}(M \cup \{\bar {b}_1 \})$ and $H^{N}(M \cup \{ \bar {b}_2 \})$ are not in K, we can not simply apply the amalgamation property and instead we have to do a similar argument to that of Lemma 3.5.

Given $ i \in \{1, 2\}$ , $H^{N}(M \cup \{\bar {b}_i\})$ is pure-injective so there is $L_i$ such that $N=H^{N}(M \cup \{\bar {b}_i\}) \oplus L_i$ . Let $L= (H^{N}(M \cup \{\bar {b}_2\}) \oplus L_1) \oplus (H^{N}(M \cup \{\bar {b}_2\}) \oplus L_2)$ . Observe that $L \in K$ by closure under direct sums and the fact that $H^{N}(M \cup \{\bar {b}_1\}) \oplus L_1$ is isomorphic to $H^{N}(M \cup \{\bar {b}_2\}) \oplus L_1$ . Define $f: N=H^{N}(M \cup \{\bar {b}_1\}) \oplus L_1 \to L$ by $f(s + l_1)=(h(s), l_1, h(s), 0)$ for $s \in H^{N}(M \cup \{\bar {b}_1\})$ and $l_1 \in L_1$ . Define $g: N=H^{N}(M \cup \{\bar {b}_2\}) \oplus L_2 \to L$ by $g(q + l_2)=(q, 0, q, l_2)$ for $q \in H^{N}(M \cup \{\bar {b}_2\})$ and $l_2 \in L_2$ . It is easy to check that $L, f, g$ witness that $\mathbf {gtp}(\bar {b}_{1}/M; N_1) = \mathbf {gtp}(\bar {b}_{2}/M; N_2)$ .⊣

Proof. Fix $\{ M_i : i < \alpha \}$ a witness to the fact that M is a $(\lambda , \alpha )$ -limit model. We show that every $p(x) \ M$ -consistent $pp$ -type over $A \subseteq M$ with $|A| \leq |R| + \aleph _0$ is realized in M.Footnote ⁴ This enough to show that M is pure-injective by [Reference Prest29, 2.8].

Observe that p is a $PE(M)$ -consistent $pp$ -type as $M \preceq PE(M)$ . Since $PE(M)$ is pure-injective, it is saturated for $pp$ -types [Reference Prest29, 2.8], so there is $a \in PE(M)$ realizing p. As $\operatorname {\mathrm {cf}}(\alpha )\geq (|R| + \aleph _0)^+$ , there is $i < \alpha $ such that $A \subseteq M_i$ . Applying downward Löwenheim–Skolem to $M_i \cup \{ a \}$ in $PE(M)$ we obtain $N \in \mathbf {K}_\lambda $ with $M_i \leq _p N$ and $a \in N$ . Then there is $f: N \xrightarrow [M_i]{} M$ because $M_{i + 1}$ is universal over $M_i$ . Hence $f(a) \in M$ realizes p.⊣

Notation 3.17. For $\mathbf {K}$ satisfying Hypothesis 3.1, let $\tilde {M}_{\mathbf {K}}$ be the $(2^{\operatorname {LS}(\mathbf {K})}, \omega )$ -limit model of $\mathbf {K}$ and $\tilde {T}_{\mathbf {K}}=Th(\tilde {M}_{\mathbf {K}})$ .

Proof. (1) $\Rightarrow $ (2) Clear.

(2) $\Rightarrow $ (3) Let $\lambda \geq \operatorname {LS}(\mathbf {K})^+$ such that $\mathbf {K}$ has uniqueness of limit models of size $\lambda $ . Let M be a $(\lambda , (|R| + \aleph _0)^+)$ -limit model in $\mathbf {K}$ . It follows from Lemma 3.14 that $M^{(\aleph _0)}$ is the $(\lambda , \omega )$ -limit model. As $\mathbf {K}$ has uniqueness of limit models of size $\lambda $ , we have that M is isomorphic to $M^{(\aleph _0)}$ . Since M is pure-injective by Lemma 3.13, it follows that $M^{(\aleph _0)}$ is pure-injective. Hence M is $\Sigma $ -pure-injective. Since limit models are elementarily equivalent by Lemma 3.15 and $\Sigma $ -pure-injectivity is preserved under elementarily equivalence by Fact 2.7, it follows that every limit model is $\Sigma $ -pure-injective.

(3) $\Rightarrow $ (4) Let $N\in \mathbf {K}$ and $N'$ be a $(\|N\|^{|R| + \aleph _0}, \omega )$ -limit model, this exists by Theorem 3.11. Then there is $f: N \to N'$ a pure embedding by [Reference Mazari-Armida24, 2.10]. Since $N'$ is $\Sigma $ -pure-injective and f is a pure embedding, it follows from Fact 2.7 that N is $\Sigma $ -pure-injective. Hence every model in $\mathbf {K}$ is pure-injective.

(4) $\Rightarrow $ (5) Let M be a $(2^{\operatorname {LS}(\mathbf {K})}, \omega )$ -limit model. By (4) and closure under direct sums we have that M is $\Sigma $ -pure-injective, so $Th(M)$ is $\lambda $ -stable for every $\lambda \geq |R| + \aleph _0$ by Fact 2.7. As $Th(M)=\tilde {T}_{\mathbf {K}}$ by definition, it follows from Lemma 3.18 that $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ . Therefore, by [Reference Shelah44, Section II.1.16] there exist a $\lambda $ -limit model for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ .

Regarding uniqueness, observe that given M and $N \ \lambda $ -limit models, there are $f: M \to N$ and $g: N \to M$ pure embeddings by [Reference Mazari-Armida24, 2.10]. Since we have that M and N are pure-injective, it follows from Fact 2.6 that M and N are isomorphic.

(5) $\Rightarrow $ (1) Clear.⊣

Fact 3.22 [Reference Prest30, 4.4.17].

Let R be a ring. The following are equivalent.

1. (1) R is left noetherian.

2. (2) The class of absolutely pure left R-modules is the same as the class of injective left R-modules.

3. (3) Every direct sum of injective left R-modules is injective.

Proof. Recall that absolutely pure modules and locally injective modules satisfy Hypothesis 3.1, so we can use the results from the previous subsection. More precisely, we use Theorem 3.20.(4) to show the equivalences.

(1) $\Rightarrow $ (2) If R is noetherian, then every absolutely pure module is injective by Fact 3.22. Hence, every absolutely pure module is pure-injective. So the result follows from Theorem 3.20.

(2) $\Rightarrow $ (3) Every locally injective module is absolutely pure by [Reference Ramamurthi and Rangaswamy32, 3.1]. Then it follows that every locally injective module is pure-injective by (2). Hence, the class of locally injective R-modules is superstable.

(3) $\Rightarrow $ (1) We show that the direct sum of injective modules is injective, this is enough by Fact 3.22. Let $\{ M_i : i \in I \}$ be a family of injective modules. As they are all locally injective, we have that $\bigoplus _{i \in I} M_i$ is locally injective. Moreover, as $(R\text {-l-inj}, \leq _p)$ is superstable, we have that $\bigoplus _{i \in I} M_i$ is also pure-injective by Theorem 3.20. Recall that locally injective modules are absolutely pure, so $\bigoplus _{i \in I} M_i$ is absolutely pure and pure-injective. Therefore, $\bigoplus_{i \in I} M_i$ is injective. Hence R is noetherian.⊣

Proof. If $(R\text {-Inj}, \leq _{p})$ is an AEC then the direct sum of injective modules is an injective module because injective modules are closed under finite direct sums. Hence R is left noetherian. On the other hand, if R is left noetherian, then injective modules are the same as absolutely pure modules by Fact 3.22. Hence $(R\text {-Inj}, \leq _{p})$ is an AEC.

The moreover part follows directly from Theorem 3.23.⊣

Proof.

1. (1) Since $(R\text {-AbsP}, \leq _{p})$ is superstable, then by Theorem 3.23 R is left noetherian. Then R is left coherent, so it follows from [Reference Prest30, 3.4.24] that absolutely pure modules are first-order axiomatizable

2. (2) The proof is similar to that of (1), using that if R is noetherian then the class of absolutely pure modules is the same as the class of locally injective modules.⊣

Proof. Recall that R-modules and locally pure-injective R-modules satisfy Hypothesis 3.1. We use Theorem 3.20.(4) to show the equivalences. The equivalence between (1) and (2) and the direction (2) to (3) are straightforward. We show (3) to (1).

Let M be an R-module, then $PE(M)$ is locally pure-injective and $M \leq _p PE(M)$ . Observe that $PE(M)^{(\aleph _0)}$ is locally pure-injective. Then $PE(M)^{(\aleph _0)}$ is pure-injective by hypothesis (3), so $PE(M)$ is $\Sigma $ -pure-injective. Hence, M is pure-injective by Fact 2.7. Therefore, R is left pure-semisimple.⊣

Proof. If $(R\text {-pi}, \leq _{p})$ is an AEC, then $M^{(\aleph _0)}=\bigcup _{n < \omega } M^{n}$ is pure-injective for every pure-injective module M as pure-injective modules are closed under finite direct sums. So every pure-injective module is $\Sigma $ -pure-injective. Then doing an argument similar to that of the previous result, one can show that R is left pure-semisimple. On the other hand, if R is left pure-semisimple, then all modules are pure-injective. Hence $(R\text {-pi}, \leq _{p})$ is an AEC.

The moreover part follows directly from Theorem 3.26.⊣

Proof. Since $(R\text {-l-pi}, \leq _{p})$ is superstable, then by Theorem 3.26 R is left pure-semisimple. Hence, every R-module is pure-injective, so in particular locally pure-injective.⊣

Proof. The backward direction follows from Theorem 3.26 as $(R\text {-Mod}, \leq _p)$ satisfies the hypothesis of (2). We show the forward direction.

Let $\mathbf {K}$ be a class satisfying the hypothesis of (2). Then $\mathbf {K}$ is closed under pure-injective envelopes as every module is pure-injective since we are assuming that the ring is left pure-semisimple. Hence, $\mathbf {K}$ satisfies Hypothesis 3.1. Therefore, $\mathbf {K}$ is superstable by Theorem 3.20.(4) and the hypothesis on the ring.⊣

Proof.

1. (1) $\Rightarrow $ : Since R is a Dedekind domain, every divisible R-module is injective by [Reference Rotman35, 4.24]. As injective modules are pure-injective, $(R\text {-Div}, \leq _p)$ is superstable by Theorem 3.20.

   $\Leftarrow $ : Recall that a module is h-divisible if it is the epimorphic image of an injective module. Therefore, the class of h-divisible R-modules is contained in the class of divisible R-modules. Then every h-divisible R-module is pure-injective by Theorem 3.20. Therefore, R is a Dedekind domain by [Reference Salce37, 2.5].

2. (2) $\Rightarrow $ : If R is a field, clearly R is a Prüfer domain. So the class of flat modules is the same as the class of torsion-free modules by [Reference Rotman35, 4.35]. Then $(R\text {-TF}, \leq _p)$ is superstable since R is perfect and by [Reference Mazari-Armida27, 3.15].

   $\Leftarrow $ : It follows from Theorem 3.20 and [Reference Salce37, 2.3] that R is a Prüfer domain. So, as before, the class of flat modules is the same as the class of torsion-free modules. Then R is left perfect by [Reference Mazari-Armida27, 3.15]. Therefore, R is a field by [Reference Salce38, 2.3].⊣

Proof. $\Rightarrow $ : Assume that the following is a pure-exact sequence:

with $B \in K$ . As $A \leq _p B$ and K is closed under pure submodules, it follows that $A \in K$ . Then by hypothesis we have $(B \oplus B)/ \{ (a, -a) : a \in A \} \in K$ because this is the pushout of $(A \hookrightarrow B, A \hookrightarrow B)$ .

Define $f: B/A \to (B \oplus B)/ \{ (a, -a) : a \in A \} $ by $f( b + A) = (b, -b) + \{ (a, -a) : a \in A \}$ . It is easy to check that f is a pure embeddings. As K is closed under pure submodules, this implies that $B/A \in K$ . Hence $C \in K$ .

$\Leftarrow $ : Let $A \leq _p B, C$ be a span with $A,B, C \in K$ . Observe that $(B \oplus C)/ \{ (a, -a) : a \in A \}$ is the pushout of $(A \hookrightarrow B, A \hookrightarrow C)$ . Since K is closed under direct sums $B \oplus C \in K$ and it is straightforward to show that $\pi : B \oplus C \to (B \oplus C)/ \{ (a, -a) : a \in A \}$ is a pure epimorphism. Therefore, $(B \oplus C)/ \{ (a, -a) : a \in A \} \in K$ .⊣

Fact 4.12 [Reference Lieberman, Rosický and Vasey21, 2.7].

If $\mathbf {K}$ satisfies Hypothesis 4.1, then is a weakly stable independence relation.

Notation 4.13. Given an independence relation on an AEC, recall that one writes if $M \le _{\mathbf {K}} M_1, M_2 \le _{\mathbf {K}} N$ and where $i_1: M \to M_1, i_2: M \to M_2,j_1: M_1 \to N, j_2: M_2 \to N$ are the inclusion maps.

Proof. Let $M_1, M_2 \leq _p N$ . We build two increasing continuous chains $\{M_{0,i} : i < \omega \}$ and $\{M^{\prime }_{1,i} : i < \omega \}$ such that:

1. (1) $M^{\prime }_{1,0} = M_1$ .

2. (2) $M_{0, i} \leq _p M^{\prime }_{1, i+1}, M_2 \leq _p N$ .

3. (3) $\| M_{0, i} \|, \| M^{\prime }_{1,i} \| \leq \| M_1 \| + |R| + \aleph _0$ .

4. (4) If $\bar {a} \in M^{\prime }_{1,i}$ , $\phi (\bar {x}, \bar {y})$ is a $pp$ -formula and there is $\bar {m} \in M_2$ such that $N \vDash \phi (\bar {a}, \bar {m})$ , then there is $\bar {l}\in M_{0, i}$ such that $N \vDash \phi (\bar {a}, \bar {l})$ .

Construction. $Base$ : Let $M^{\prime }_{1,0} = M_1$ . For each $\bar {a} \in M_1$ and $\phi (\bar {x}, \bar {y})$ a $pp$ -formula, if there is $\bar {m} \in M_2$ such that $N \vDash \phi (\bar {a}, \bar {m})$ let $\bar {m}_\phi ^{\bar {a}}$ be a witness in $M_2$ and $\bar {0}$ otherwise. Let $M_{0,0}$ be the structure obtained by applying Downward Löwenheim–Skolem to $\bigcup \{ \bar {m}_\phi ^{\bar {a}} : \bar {a} \in M_1 \text { and } \phi \text { is a }pp\text {-formula} \}$ in $M_2$ . It is easy to see that $M_{0,0}$ satisfies what is needed.

$Induction\ step$ : Let $M^{\prime }_{1, i+1}$ be the structured obtained by applying Downward Löwenheim–Skolem to $M_{0,i} \cup M^{\prime }_{1, i}$ in N. Construct $M_{0, i +1}$ as we constructed $M_{0,0}$ , but replacing $M^{\prime }_{1,0}$ by $M^{\prime }_{1,i+1}$ and making sure that $M_{0, i} \leq _p M_{0, i +1}$ .

Enough. Let $M_0= \bigcup _{ i < \omega } M_{0, i}$ and $M^{\prime }_1 = \bigcup _{i < \omega } M^{\prime }_{1,i}$ . Observe that $\| M_0 \| \leq \| M_1 \| + |R| + \aleph _0$ and we show that .

Recall that the pushout in R-Mod is given by:

Moreover, $t: (M^{\prime }_1 \oplus M_2)/ \{ (m, -m) : m \in M_0 \} \to N$ is given by $t([(m, n)])= m + n$ . So we are left to show that t is a pure embedding.

We begin by proving that t is injective, so assume that $m_1 + n_1 = m_2 + n_2$ with $m_i \in M^{\prime }_1$ and $n_i \in M_2$ for $i \in \{1, 2\}$ . Then $N \vDash m_1-m_2 = (n_2 -n_1)$ , so by condition (4) of the construction there is $m \in M_0$ such that $N \vDash m_1 - m_2 = m$ . Hence $[(m_1, n_1)] = [(m_2, n_2)]$ in the pushout.

We show that t is pure. Let $\phi (y)$ be a $pp$ -formula such that $N \vDash \phi (m+ n)$ with $m \in M_1'$ and $n \in M_2$ . So $N\vDash \exists w ( \phi (w) \wedge w = z + z') (m, n)$ . Observe that this is a $pp$ -formula, $m \in M^{\prime }_1$ and $n \in M_2$ , then by condition (4) of the construction there is $p \in M_0$ such that $N\vDash \exists w ( \phi (w) \wedge w = z + z') (m, p)$ . So $N \vDash \phi (m + p)$ . Assume $\phi (y)$ is equal to $\exists \bar {x} \theta (\bar {x}, y)$ for $\theta (\bar {x}, y)$ quantifier-free formula. Then as $M^{\prime }_1 \leq _p N$ there is $\bar {m}^\star \in M^{\prime }_1$ such that

(1) $$ \begin{align} N \vDash \theta(\bar{m}^\star, m + p). \end{align} $$

As solutions to $pp$ -formulas form a subgroup, it is easy to get that $N \vDash \phi (n-p)$ . Then as $M_2 \leq _p N$ there is $\bar {n}^\star \in M_2$ such that

(2) $$ \begin{align} N \vDash \theta(\bar{n}^\star, n - p). \end{align} $$

So by adding equation (1) and (2) we obtain that:

(3) $$ \begin{align} N \vDash \theta(\bar{m}^\star + \bar{n}^\star, m + n). \end{align} $$

Therefore, $t: (M^{\prime }_1 \oplus M_2)/ \{ (m, -m) : m \in M_0 \} \to N$ is a pure embedding.⊣

Proof. (1) follows from Fact 4.12 and [Reference Lieberman, Rosický and Vasey20, 8.5]. As for (2), this follows from Theorem 4.14.⊣

Proof. Let $M \in \mathbf {K}_\lambda $ with $\lambda ^{|R| + \aleph _0}=\lambda $ . Assume for the sake of contradiction that $| \mathbf {gS}(M) |> \lambda $ and let $\{ p_i : i < \lambda ^+ \}$ be an enumerations without repetitions of types in $\mathbf {gS}(M)$ .

By Lemma 4.16, for every $ i < \lambda ^+$ , there is $N_i \leq _p M$ such that $p_i$ does not fork over $N_i$ and $\| N_i \| = |R| + \aleph _0$ . Then by the pigeon hole principle and using that $\lambda ^{|R| + \aleph _0}=\lambda $ , we may assume that there is an $N \in K$ such that $N_i = N$ for every $i < \lambda ^+$ . Therefore, by uniqueness, there are $i \neq j < \lambda ^+$ such that $p_i = p_j$ . This is clearly a contradiction.⊣

Proof. Follows from Lemma 4.16 and [Reference Lieberman, Rosický and Vasey20, 8.16].⊣

Proof. By Corollary 3.10 we have $\mathbf {K}$ is fully $(< \aleph _0)$ -tame. Then it follows from [Reference Lieberman, Rosický and Vasey20, 8.8, 8.9] that has the $(<\aleph _0)$ -witness property. The moreover part follows from Fact 4.12 and Theorem 4.14.⊣

Question 4.23. If $\mathbf {K}$ satisfy Hypothesis 4.1, is a stable independence relation?

Proof. Observe that the class of left R-modules with pure embeddings satisfies Hypotheses 3.1 and 4.1, then by Lemma 4.22 is a stable independence relation. Since R-Mod with pure embeddings is an accessible cellular category which is retract-closed, coherent and $\aleph _0$ -continuous. Therefore, pure embeddings are cofibrantly generated by [Reference Lieberman, Rosický and Vasey21, 3.1].⊣

Proof. The backward direction is obvious so we prove the forward direction. Since $\mathbf {K}$ admits intersection, by [Reference Vasey46, 2.18], there is $f: cl^{N_1}_{\mathbf {K}}(\bar {a} \cup A) \cong _M cl^{N_2}_{\mathbf {K}}(\bar {b} \cup A)$ with $f(\bar {a})=\bar {b}$ . Then using the proposition above we have that $cl^{N_1}_{\mathbf {K}}(\bar {a} \cup A) = cl^{N_1}_{\mathbf {K}^\star }(\bar {a} \cup A)$ and $cl^{N_2}_{\mathbf {K}}(\bar {b} \cup A)= cl^{N_2}_{\mathbf {K}^\star }(\bar {b} \cup A)$ . So the result follows from the fact that $\mathbf {K}^\star $ admits intersections and [Reference Vasey46, 2.18].⊣

Proof. We show that $\mathbf {K}$ is closed below $(R\text {-Mod}, \leq _p)$ . Observe that the only things that need to be shown are that $(R\text {-Mod}, \leq _p)$ admits intersections and that K is closed under pure submodules. This is the case as every module is absolutely pure by the hypothesis on the ring.⊣