Proof $(a) \Rightarrow (b)$ is clear.

$(b) \Rightarrow (d)$ . Consider the exact sequence $0\rightarrow N' \rightarrow I_0 \overset {t}{\rightarrow } I_1$ , where $I_0$ and $I_1$ are injective modules in SMod. Then the sequence $0\rightarrow N \overset {h}{\rightarrow } R\mathrm {Hom}_S(G_P,I_0) \overset {g}{\rightarrow } R\mathrm {Hom}_S(G_P,I_1)$ is exact, where $g=R\mathrm {Hom}_S(G_P,t)$ . This implies that the following sequence is also exact

$$ \begin{align*}0\longrightarrow S\mathrm{Hom}_R(P,N) &\overset{S\mathrm{Hom}_R(P,h)}{\longrightarrow} S\mathrm{Hom}_R(P,R\mathrm{Hom}_S(G_P,I_0)) \\ &\overset{S\mathrm{Hom}_R(P,g)}{\longrightarrow} S\mathrm{Hom}_R(P,R\mathrm{ Hom}_S(G_P,I_1)).\end{align*} $$

We know that by [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.1], the pair of functors

$$ \begin{align*} G_P\otimes_R-:R\mathrm{Mod} \rightarrow S\mathrm{Mod}\ \text{and}\ R\mathrm{Hom}_S(G_P,-):S\mathrm{Mod} \rightarrow R\mathrm{Mod,} \end{align*} $$

is adjoint via the isomorphisms (for $X \in R$ Mod and $Y \in S$ Mod)

$$ \begin{align*} \psi_{X,Y}:\mathrm{Hom}_R(X,R\mathrm{Hom}_S(G_P,Y)) &\rightarrow \mathrm{Hom}_S(G_P\otimes_RX,Y) \text{via} \\ \beta& \mapsto \left[ \sum_{i=1}^t\alpha_i\otimes x_i\mapsto \sum_{i=1}^t(\alpha_i)(x_i)\beta \right]. \end{align*} $$

Also, by [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.8], for each $Y \in S$ Mod there exists an S-module isomorphism

$$ \begin{align*}\nu_Y:Y \rightarrow S\mathrm{Hom}_R(P,R\mathrm{Hom}_S(G_P,Y)),\end{align*} $$

which determines a natural isomorphism $\mathrm {id}_{S\mathrm {Mod}} \rightarrow S\mathrm {Hom}_R(P,R\mathrm {Hom}_S(G_P,-))$ . Moreover by the proof of [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.5], there exists an S-module isomorphism

$$ \begin{align*}\eta_M: G_P \otimes_R M \rightarrow S\mathrm{Hom}_R(P,M).\end{align*} $$

Set $I^{\prime }_i:=S\mathrm {Hom}_R(P,R\mathrm {Hom}_S(G_P,I_i))$ and $\delta _i:= \mathrm {Hom}_R(M,R\mathrm {Hom}_S(G_P,\nu _{I_i}))$ , for $i=0,1$ and let M be a unitary left R-module. Therefore, the following diagram is commutative

where $M'=S\mathrm {Hom}_R(P,M)$ , $N'=S\mathrm {Hom}_R(P,N)$ , $g'=S\mathrm {Hom}_R(P,g)$ , $h'=S\mathrm {Hom}_R(P,h)$ , $g"'=\mathrm {Hom}_R(M,R\mathrm {Hom}_S(G_P,g'))$ , $h"=\mathrm {Hom}_S(S\mathrm {Hom}_R(P,M),h')$ and $g"=\mathrm { Hom}_S(S\mathrm {Hom}_R(P,M),g')$ . By using the proof of [Reference Fazelpour and Nasr-Isfahani16, Lemmas 2.5 and 2.8] and the five lemma, we can see that the functor $S\mathrm {Hom}_R(P,-)$ gives the following isomorphism:

$$ \begin{align*}\mathrm{Hom}_R(M,N) \rightarrow \mathrm{Hom}_S(S\mathrm{Hom}_R(P,M),S\mathrm{Hom}_R(P,N)).\end{align*} $$

$(d) \Rightarrow (a).$ Put $M=R$ and then we can easily see this isomorphism induces asked isomorphism.

$(b) \Rightarrow (c)$ is clear.

$(c) \Rightarrow (a).$ It follows from $(b) \Leftrightarrow (a)$ and the five lemma.

Proof $(a) \Rightarrow (b)$ . Since $\mathrm {Soc}(E) \in \mathrm {Add}(\bigoplus _{i=1}^nS_i)$ , $st_P(\mathrm {Soc}(E))=0$ . Then $st_P(E)=0$ because $\mathrm {Soc}(E)$ is an essential submodule of E and $st_P(E) \cap \mathrm {Soc}(E) \subseteq st_P(\mathrm {Soc}(E))$ .

$(b)\Rightarrow (a)$ . Assume that there exists a simple direct summand $S'$ of $\mathrm {Soc}(E)$ such that $S' \ncong S_i$ for each $i~(1 \leq i \leq n)$ . Then $st_P(S') \neq 0$ since otherwise there is a nonzero morphism $g \in \mathrm {Hom}_R(P,S')$ . Then there exists a nonzero R-module homomorphism $h: P \rightarrow P(S')$ such that $hf=g$ , where $f:P(S') \rightarrow S'$ is a projective cover of $S'$ in RMod. Hence, $S' \cong S_j$ for some j, which is a contradiction. Therefore, $st_P(S') \neq 0$ and so $st_P(E) \neq 0,$ which is a contradiction. Consequence $\mathrm {Soc}(E) \in \mathrm {Add}(\bigoplus _{i=1}^nS_i)$ .

$(b)\Rightarrow (c)$ . It follows from [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.9].

$(c)\Rightarrow (b)$ . From E is injectively copresented over P, by Proposition 2.1(d) we get

$$ \begin{align*}\mathrm{Hom}_R(st_P(E),E) \cong \mathrm{Hom}_S(\mathrm{Hom}_R(P,st_P(E)),\mathrm{Hom}_R(P,E)),\end{align*} $$

as abelian groups. Since $\mathrm {Hom}_R(P,st_P(E))=0$ , $\mathrm {Hom}_R(st_P(E),E)=0$ and so $st_P(E)=0$ .

Proof It follows from [Reference Wisbauer48, Propositions 32.4 and 51.6] and the fact that $\widehat {\mathrm {{Hom}}}_{\Lambda }(V,V_{\alpha }) \cong Re_{\alpha }$ as R-modules, where $e_{\alpha }=\pi _{\alpha } \varepsilon _{\alpha }$ , $\pi _{\alpha }:U \rightarrow U_{\alpha }$ is the canonical projection and $\varepsilon _{\alpha }:U_{\alpha } \rightarrow U$ is the canonical injection.

$(a)$ . It follows from [Reference Fazelpour and Nasr-Isfahani15, Lemma 2.3].

$(b)$ . Since V is a generator in $\Lambda $ -Mod, by [Reference Wisbauer48, Proposition 51.7], the functor $\widehat {\mathrm {{Hom}}}_{\Lambda }(V,-):\Lambda $ -Mod $\rightarrow R$ Mod preserves indecomposable modules, injective modules, and essential extensions. Hence, there exists a minimal injective resolution $0 \longrightarrow {\widehat {\mathrm {{Hom}}}}_{\Lambda }(V,V_{\alpha }) \longrightarrow {\widehat {\mathrm {{Hom}}}}_{\Lambda }(V,E(V_{\alpha })) \longrightarrow Y$ , where Y is a direct summand of ${\widehat {\mathrm {{Hom}}}}_{\Lambda }(V,E(E(V_{\alpha })/V_{\alpha }))$ . Let $\lbrace \Lambda e_1/J(\Lambda )e_1, \cdots ,\Lambda e_m/J(\Lambda )e_m \rbrace $ be a complete set of nonisomorphic simple left $\Lambda $ -modules, where $\lbrace e_1, \cdots , e_m \rbrace $ is a basic set of idempotents of $\Lambda $ . Since $\Lambda $ is a left artinian ring, the injective hull of any finitely generated left $\Lambda $ -module belongs to be $\mathrm {add}(\bigoplus _{i=1}^mE(\Lambda e_i/J(\Lambda )e_i))$ . On the other hand, since V is a cogenerator in $\Lambda $ -Mod, each $E(\Lambda e_l/J(\Lambda )e_l) \in \mathrm {add}(V)$ . Since the functor $\widehat {\mathrm {{Hom}}}_{\Lambda }(V,-)$ preserves direct sums, ${\widehat {\mathrm {{Hom}}}}_{\Lambda }(V,E(V_{\alpha }))$ and Y are finitely generated projective unitary left R-modules. Hence, R has the property $(iv)$ .

Now we show that R has the property $(iii)$ . Since V is a generator, $\Lambda e_i \cong V_{\alpha _i}$ for some $\alpha _i \in J$ . Let $\rho _i:\Lambda e_i \rightarrow \Lambda e_i/J(\Lambda )e_i$ be a projective cover of $\Lambda e_i/J(\Lambda )e_i$ and $\pi _{\alpha _i}: V \rightarrow V_{\alpha _i}$ be the canonical projection. Then $\pi _{\alpha _i} \rho _{\alpha _i} \rho _i$ is a nonzero morphism in $\widehat {\mathrm {{Hom}}}_{\Lambda }(V,\Lambda e_i/J(\Lambda )e_i)$ , where $\rho _{\alpha _i}: V_{\alpha _i} \rightarrow \Lambda e_i$ is an isomorphism. Consider the R-module homomorphism

$$ \begin{align*}\varphi:=\widehat{\mathrm{{Hom}}}_{\Lambda}(V,\pi_{\alpha_i} \rho_{\alpha_i} \rho_i): \widehat{\mathrm{{End}}}_{\Lambda}(V) \rightarrow \widehat{\mathrm{{Hom}}}_{\Lambda}(V,\Lambda e_i/J(\Lambda)e_i).\end{align*} $$

It is easy to see that $\varphi $ is a nonzero morphism. Also $J(R)\widehat {\mathrm {{End}}}_{\Lambda }(V) \subseteq \mathrm {Ker}\varphi $ since otherwise there exist $t \in J(R)$ and $\gamma \in \widehat {\mathrm {{End}}}_{\Lambda }(V)$ such that $t\gamma \pi _{\alpha _i} \rho _{\alpha _i} \rho _i \neq 0$ . It follows that $\mathrm {Im}~t\gamma \pi _{\alpha _i} \rho _{\alpha _i} \nsubseteq \mathrm {Ker}\rho _i$ . Since $\rho _i: \Lambda e_i \rightarrow \Lambda e_i/J(\Lambda )e_i$ is a projective cover of $\Lambda e_i/J(\Lambda )e_i$ , the morphism $t\gamma \pi _{\alpha _i} \rho _{\alpha _i}: V \rightarrow \Lambda e_i$ is an epimorphism and so there is a $\Lambda $ -module homomorphism $h: \Lambda e_i \rightarrow V$ such that $ht\gamma \pi _{\alpha _i} \rho _{\alpha _i} = id_{\Lambda e_i}$ . Let $\varepsilon _{\alpha _i}: V_{\alpha _i} \rightarrow V$ be the canonical injection. Since $\pi _{\alpha _i} \rho _{\alpha _i} ht\gamma \pi _{\alpha _i} \rho _{\alpha _i} {\rho ^{-1}_{\alpha _i}} \varepsilon _{\alpha _i}= \pi _{\alpha _i} \varepsilon _{\alpha _i}$ , $\pi _{\alpha _i}\varepsilon _{\alpha _i} \in J(R)$ . By [Reference Wisbauer48, Proposition 49.7], this is a contradiction with the fact that $J(R)$ contains no idempotent. Consequently, $\mathrm {Im}\varphi $ is a nonzero semisimple submodule of $\widehat {\mathrm {{Hom}}}_{\Lambda }(V,\Lambda e_i/J(\Lambda )e_i)$ . It follows that $\mathrm { Soc}(\widehat {\mathrm {{Hom}}}_{\Lambda }(V,E(\Lambda e_i/J(\Lambda )e_i))) \neq 0$ and so ${\widehat {\mathrm {{Hom}}}_{\Lambda }(V,E(\Lambda e_i/J(\Lambda )e_i))}$ has a simple essential socle. Hence,

$$ \begin{align*}\widehat{\mathrm{{Hom}}}_{\Lambda}(V,E(\Lambda e_i/J(\Lambda)e_i)) \cong E(R\pi_{\alpha_i} \rho_{\alpha_i} \rho_i),\end{align*} $$

as R-modules. Since $R\pi _{\alpha _i} \rho _{\alpha _i} \rho _i \cong R\pi _{\alpha _j} \rho _{\alpha _j} \rho _j$ if and only if $\Lambda e_i/J(\Lambda )e_i \cong \Lambda e_j/J(\Lambda )e_j$ , so $\lbrace R\pi _{\alpha _i} \rho _{\alpha _i} \rho _i ~|~1 \leq i \leq m \rbrace $ is a set of nonisomorphic simple unitary left R-modules such that $\mathrm {add}(\bigoplus _{i=1}^m E(R\pi _{\alpha _i} \rho _{\alpha _i} \rho _i)) \subseteq \mathrm {proj}(R) \cap \mathrm {inj}(R)$ . But because for each finitely generated indecomposable projective unitary left R-module $Q, Q \in \mathrm {add}(\bigoplus _{i=1}^m E(R\pi _{\alpha _i} \rho _{\alpha _i} \rho _i))$ when $Q \in \mathrm {inj}(R)$ , then $\mathrm {proj}(R) \cap \mathrm {inj}(R) \subseteq \mathrm { add}(\bigoplus _{i=1}^m E(R\pi _{\alpha _i} \rho _{\alpha _i} \rho _i))$ and so R has the property $(iii)$ .

We know that the mapping $R \pi _{\alpha _i}\rho _{\alpha _i} \rightarrow R\pi _{\alpha _i}\rho _{\alpha _i} \rho _i$ defined by $r\pi _{\alpha _i}\rho _{\alpha _i} \longmapsto r\pi _{\alpha _i} \rho _{\alpha _i} \rho _i$ is a nonzero R-module epimorphism. Since $R\pi _{\alpha _i}\rho _{\alpha _i} = \widehat {\mathrm {{Hom}}}_{\Lambda }(V,\Lambda e_i)$ , $R\pi _{\alpha _i}\rho _{\alpha _i}$ is a finitely generated indecomposable projective unitary left R-module. So it is a projective cover of $R\pi _{\alpha _i} \rho _{\alpha _i} \rho _i$ . Set $e=e_1 + \cdots + e_m$ . Since $P= \bigoplus _{i=1}^m\widehat {\mathrm {{Hom}}}_{\Lambda }(V,\Lambda e_i) \cong \widehat {\mathrm {{Hom}}}_{\Lambda }(V,\Lambda e)$ as R-modules and $\Lambda e \in \mathrm {add}(V)$ by (3.1), we have

$$ \begin{align*} S= \mathrm{End}_R(P) \cong \mathrm{End}_R({{\widehat{\mathrm{Hom}}}}_{\Lambda}(V,\Lambda e)) \cong \mathrm{End}_{\Lambda}(\Lambda e) \cong e\Lambda e, \end{align*} $$

as rings. It follows that S is Morita equivalent to $\Lambda $ and the result holds.

Proof By Proposition 3.3, $R_{\Lambda }$ has the properties $(i)$ and $(iii)$ – $(iv)$ . Also by [Reference Hullinger29, Theorem 2] and [Reference Fazelpour and Nasr-Isfahani15, Lemma 2.2], $\mathrm {l.gl.dim}R_{\Lambda } \leq 2$ . Moreover, if we had $\mathrm {l.gl.dim}R_{\Lambda } = 1$ , then every finitely generated submodule of a projective unitary left $R_{\Lambda }$ -module is projective. By [Reference Wisbauer48, Proposition 52.8], $\Lambda $ is a left semisimple ring and so $R_{\Lambda }$ is Morita equivalent to $\Lambda $ . This yields that every unitary left $R_{\Lambda }$ -module is projective, which is a contradiction. This proves that $\mathrm {l.gl.dim}R_{\Lambda }$ is either 0 or 2. Also by Proposition 3.3(a), $R_{\Lambda }$ has the properties $(ii)$ .

Proof $(a)$ . The proof is given in several steps.

Step 1. We show that S is a left artinian ring. First we show that S is a right perfect ring. We know that each $P_i \cong Re_{\alpha _i}$ as R-modules for some $\alpha _i \in J$ . Also, by the property $(iv)$ , there exists a minimal injective resolution $0 \rightarrow Re_{\alpha _i} \rightarrow E_0 \rightarrow E_1$ where $E_0$ and $E_1$ are projective and injective unitary left R-modules. Since for each $l \in \lbrace 0, 1\rbrace $ , $E_l$ is projective unitary left R-module, $E_l \cong \bigoplus _{j \in B_l}Re_{\alpha _j}$ . Since it is injective in RMod, each $Re_{\alpha _j} \in \mathrm {proj}(R) \cap \mathrm {inj}(R)$ . By the property $(iii)$ , each $Re_{\alpha _j} \in \mathrm {add}(\bigoplus _{i=1}^nE(S_i))$ and so $E_l \in \mathrm {Add}(\bigoplus _{i=1}^nE(S_i))$ . By Lemma 2.3, each $E_l \in R\mathrm {Hom}_S(G_P, S\mathrm {Mod})$ . Because $\gamma =(\gamma _M): id_{R\mathrm {Mod}} \rightarrow R\mathrm {Hom}_S(G_P, S\mathrm {Hom}_R(P,-))$ is a natural transformation and $0 \rightarrow Re_{\alpha _i} \rightarrow E_0 \rightarrow E_1$ is an exact sequence, the canonical morphism $\gamma _{Re_{\alpha _i}}: Re_{\alpha _i} \rightarrow R\mathrm {Hom}_S(G_P,S\mathrm {Hom}_R(P,Re_{\alpha _i}))$ is an isomorphism. Hence, each $P_i \in R\mathrm {Hom}_S(G_P, S\mathrm {Mod})$ . Let $\pi _i: P \rightarrow P_i$ and $\varepsilon _i: P_i \rightarrow P$ be the canonical projection and injection, respectively. Set $f_i=\pi _i\varepsilon _i$ . Then it is easy to see that $f_1, \cdots ,f_n$ are orthogonal idempotents in S with $1_S= f_1 + \cdots + f_n$ . Since $\mathrm {Hom}_R(P, P_i) \cong Sf_i$ as S-modules, $f_iSf_i \cong \mathrm {End}_S(\mathrm {Hom}_R(P, P_i))$ . For each i, $P_i \in R\mathrm {Hom}_S(G_P, S\mathrm {Mod})$ and hence by Corollary 2.2, $\mathrm {End}_R(P_i) \cong f_iSf_i$ as rings. Therefore, by the property $(i)$ , each $f_iSf_i$ is a right perfect ring. By [Reference Wisbauer48, Example 43.12(2)], S is a right perfect ring. Hence, by [Reference Wisbauer48, Proposition 43.9], the descending chain condition for cyclic left ideals of S holds. It follows by [Reference Wisbauer48, Proposition 31.8] that the descending chain condition for finitely generated left ideals of S also holds. This implies that S is a left artinian ring if S is left noetherian. Therefore, it is enough to show that S is a left noetherian ring. Consider the ascending chain $I_1 \subseteq I_2 \subseteq \cdots $ of left ideals of S. Then we have an ascending chain $PI_1 \subseteq PI_2 \subseteq \cdots $ of R-submodules of P. Since R is a left locally noetherian ring and P is a finitely generated unitary left R-module, the chain of submodules of P becomes stationary after finitely many steps. Since P is a finitely generated projective unitary left R-module, by [Reference Wisbauer48, Proposition 18.4(3)], we can see that the chain $I_1 \subseteq I_2 \subseteq \cdots $ becomes stationary after finitely many steps. This yields that S is a left noetherian ring.

Step 2. Let $S'$ be a simple unitary left S-module. We show that $\mathrm {Soc}(R\mathrm {Hom}_S(G_P,E(S')))$ is nonzero. Let $g: \mathrm {Hom}_R(P, Re_{\alpha _i}) \rightarrow S'$ be a projective cover of $S'$ . By using [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.8], it is not difficult to see that $\varphi := R\mathrm {Hom}_S(G_P,\mathrm {Hom}_R(P,\pi _i))R\mathrm {Hom}_S(G_P,g)$ is a nonzero morphism, where $\pi _i:R \rightarrow Re_{\alpha _i}$ is the canonical projection. We can certainly assume that $J(R)R\mathrm {End}_S(G_P) \subseteq \mathrm {Ker}\varphi $ , since otherwise there exist $\delta \in R\mathrm {End}_S(G_P)$ and $r \in J(R)$ such that $(r\delta )\varphi \neq 0$ . This yields $\mathrm {Im}~r\delta \mathrm {Hom}_R(P,\pi _i) \nsubseteq \mathrm {Ker}g$ and so $r\delta \mathrm {Hom}_R(P,\pi _i)$ is a split epimorphism because $\mathrm {Ker}g$ is a maximal small submodule of $\mathrm { Hom}_R(P,Re_{\alpha _i})$ and $\mathrm {Hom}_R(P,Re_{\alpha _i})$ is a projective left S-module. Hence, there is an S-module homomorphism $h: \mathrm {Hom}_R(P,Re_{\alpha _i}) \rightarrow G_P$ such that $hr\delta \mathrm { Hom}_R(P,\pi _i)=id_{\mathrm {Hom}_R(P,Re_{\alpha _i})}$ . So $\mathrm {Hom}_R(P,\pi _i)hr\delta \mathrm {Hom}_R(P,\pi _i\varepsilon _i)=\mathrm {Hom}_R(P,\pi _i\varepsilon _i)$ , where $\varepsilon _i:Re_{\alpha _i} \rightarrow R$ is the canonical injection. On the other hand, since $R=\bigoplus _{\alpha \in J}Re_{\alpha }$ has the property $(iv)$ and R is left locally noetherian, by Lemma 2.3, we can see that R is injectively copresented over P. Hence, by Proposition 2.1, there exists a ring isomorphism $R \rightarrow R\mathrm {End}_S(G_P)$ via $s \mapsto \mathrm {Hom}_R(P, \rho _s)$ , where the mapping $\rho _s \in R\mathrm {End}_R(R)$ defined by $(x)\rho _s=xs$ for each $x \in R$ . Since $r \in J(R)$ , $\mathrm {Hom}_R(P, \rho _r) \in J(R\mathrm {End}_S(G_P))$ and so $r \delta =\mathrm {Hom}_R(P, \rho _r)\delta \in J(R\mathrm {End}_S(G_P))$ . Therefore, $\mathrm {Hom}_R(P,\pi _i\varepsilon _i) \in J(R\mathrm {End}_S(G_P))$ , which is a contradiction. Hence, $\mathrm {Im}\varphi $ is a semisimple unitary left R-module and so $\mathrm {Soc}(R\mathrm {Hom}_S(G_P,E(S')))$ is nonzero.

Step 3. Let E be an injective left S-module. We show that $R\mathrm {Hom}_S(G_P,E)$ is a projective unitary left R-module. By using [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.8] and Proposition 2.1, we can easily see that $R\mathrm {Hom}_S(G_P,E)$ is an injective module in RMod. Since R is a left locally noetherian ring, by [Reference Stenström46, Chapter V, Proposition 4.5], $R\mathrm {Hom}_S(G_P,E) = \bigoplus _{j \in A} N_j$ , where each $N_j$ is an indecomposable injective module in RMod. It is enough to show that each $N_j$ is a projective unitary left R-module. We know that each $N_j$ is injectively copresented over P. Hence, by Lemma 2.3 and the fact that R has the property $(iii)$ , it is sufficient to show that each $N_j$ has a simple essential socle. By using [Reference Fazelpour and Nasr-Isfahani16, Lemma 2.8], we can see that $E \cong \bigoplus _{j \in A}\mathrm {Hom}_R(P,N_j)$ as S-modules. Since each $N_j$ is injectively copresented over P, by Corollary 2.2 and [Reference Stenström46, Chapter V, Proposition 5.1], each $\mathrm {Hom}_R(P,N_j)$ is an indecomposable injective left S-module. By Step 1, S is a left artinian ring and so for each $j \in A$ , $\mathrm {Hom}_R(P, N_j) \cong E(S^{\prime }_j)$ as S-modules for some simple left S-module $S^{\prime }_j$ . Since $N_j \cong R\mathrm { Hom}_S(G_P, \mathrm {Hom}_R(P, N_j)) \cong R\mathrm {Hom}_S(G_P, E(S^{\prime }_j))$ as S-modules, by Step 2, $\mathrm {Soc}(N_j) \neq 0$ for each $j \in A$ . Each $N_j$ is an indecomposable injective module and hence has a simple essential socle.

Step 4. By Corollary 2.2, the functor $\mathrm {Hom}_R(P,-):R\mathrm {Hom}_S(G_P,S\mathrm {Mod}) \rightarrow S$ -Mod is an equivalence of categories. It is enough to show that $R\mathrm {Hom}_S(G_P,S\mathrm {Mod})=\mathrm {Proj}(R)$ . Let Q be a projective unitary left R-module. Since R has the properties $(i)$ and $(iv)$ , we have an exact sequence $0 \rightarrow Q \rightarrow E_0 \rightarrow E_1$ where each $E_l \in \mathrm {Proj}(R)$ . R is a left locally noetherian ring and so by [Reference Stenström46, Chapter V, Proposition 4.3], each $E_l \in \mathrm {Inj}(R)$ . Hence, by using the properties $(i)$ and $(iii)$ each $E_l \in \mathrm {Add}(\bigoplus _{i=1}^nE(S_i))$ . By using Lemma 2.3, it is not difficult to show that each $E_l$ is injectively copresented over P and so by the five lemma Q is injectively copresented over P. Now it is enough to show that each object in $R\mathrm {Hom}_S(G_P,S\mathrm {Mod})$ is a projective unitary left R-module. Let M be injectively copresented over P. Then $M \cong R\mathrm {Hom}_S(G_P, N)$ as R-modules for some left S-module N. Consider the following exact sequence:

$$ \begin{align*}0 \rightarrow M \rightarrow R\mathrm{Hom}_S(G_P, I_0) \rightarrow R\mathrm{Hom}_S(G_P, I_1),\end{align*} $$

where $0 \rightarrow N \rightarrow I_0 \rightarrow I_1$ is the minimal injective resolution of N. By Step 3, each $R\mathrm {Hom}_S(G_P, I_i)$ is a projective unitary left R-module. Since $\mathrm {l.gl.dim}(R) =0 ~\mathrm { or}~ 2$ , M is a projective unitary left R-module. Therefore, $R\mathrm {Hom}_S(G_P,S\mathrm {Mod})=\mathrm {Proj}(R)$ and the result follows.

$(b)$ . Since R has the property $(i)$ , every projective unitary left R-module is a direct sum of finitely generated unitary module with local endomorphism ring. By $(a)$ , every left S-module is a direct sum of indecomposable left S-module. Therefore, by [Reference Krause32, Corollary 2.7], S is a left pure semisimple ring.

Proof If $\Lambda $ is a left pure semisimple ring, then by Corollary 3.4, $R_{\Lambda }$ has the properties $(i)$ – $(iv)$ . Let $\Lambda $ and $\Lambda '$ be two left pure semisimple rings. If $\Lambda $ and $\Lambda '$ are Morita equivalent, then by using [Reference Fazelpour and Nasr-Isfahani16, Theorem 3.6], we can see that $R_{\Lambda }$ and $R_{\Lambda '}$ are Morita equivalent. Also, it is clear that if $R_{\Lambda }$ and $R_{\Lambda '}$ are Morita equivalent, then $\Lambda $ and $\Lambda '$ are Morita equivalent. This means that the assignment $\Lambda \mapsto R_{\Lambda }$ sends Morita equivalence classes of left pure semisimple rings to Morita equivalence classes of rings with the properties $(i)$ – $(iv)$ . Moreover, this shows that this map is injective. Now we show that this map is surjective. Let R be a ring with the properties $(i)$ – $(iv)$ . Then by Proposition 3.5, there exists an additive equivalence between $\mathrm {Proj}(R) \rightarrow S$ -Mod, where S is a left pure semisimple ring. Hence, we have an additive equivalence $\mathrm {Proj}(R) \rightarrow \mathrm {Proj}(R_{S})$ , which preserves and reflects finitely generated modules. By [Reference Fazelpour and Nasr-Isfahani16, Theorem 3.6], R and $R_S$ are Morita equivalence and so this map is surjective. Therefore, this map is a bijection between Morita equivalence classes of left pure semisimple rings and Morita equivalence classes of rings with the properties $(i)$ – $(iv)$ .

Proof $(a)$ . Since $fR$ is faithful, the canonical ring homomorphism $\rho : R \rightarrow R\mathrm {End}_{fRf}(fR)$ via $r \mapsto [\rho _r: x \mapsto xr]$ is a monomorphism. If $d \in R\mathrm {End}_{fRf}(fR)$ and $x \in fR$ such that $(x)d \neq 0$ , then $\rho _xd=\rho _{(x)d} \neq 0$ . Hence, the result holds.

$(b)$ . It is easy to see that the mapping $\varphi : R\mathrm {Hom}_{fRf}(fR, fRe) \rightarrow R\mathrm {End}_{fRf}(fR)e$ via $\alpha \mapsto \alpha \varepsilon e$ is an isomorphism of R-modules, where $\varepsilon : fRe \rightarrow fR$ is the canonical injection. Also since e is an idempotent of R and $fR$ is faithful, we can see that $Re \cong \mathrm {Im}\rho |_{Re} = \mathrm {Im}\rho e$ as R-modules. But because $R\mathrm {End}_{fRf}(fR)$ is a left quotient ring of R, $\mathrm {Im}\rho $ is an essential $\mathrm {Im}\rho $ -submodule of $R\mathrm {End}_{fRf}(fR)$ . This implies that $\mathrm {Im}\rho e$ is an essential R-submodule of $R\mathrm {End}_{fRf}(fR)e$ and so $\mathrm {Im}\rho e = R\mathrm {End}_{fRf}(fR)e$ because $Re$ is injective in RMod. Therefore, $R\mathrm {Hom}_{fRf}(fR, fRe) \cong Re$ as R-modules. Since $fR$ is a projective unitary right R-module, the functor $fR \otimes _R-$ sends R-module monomorphisms to $fRf$ -module monomorphisms. Since $Re$ is injective module in RMod and the natural transformation

$$ \begin{align*} \eta: fR \otimes_R R\mathrm{Hom}_{fRf}(fR,-) \rightarrow \text{ id via } fr \otimes \alpha \mapsto (fr)\alpha, \end{align*} $$

for $r \in R$ , $\alpha \in R\mathrm {Hom}_{fRf}(fR,X)$ , where X is a left $fRf$ -module, is an equivalence, $fRe$ is an injective left $fRf$ -module. Therefore, the result follows.

Proof $(a)$ . Since each $Re_i$ has a simple essential socle, $\mathrm {Soc}(Re_i)=Ra_i$ for some $0 \neq a_i \in R$ . Since R is a ring with local units, $a_ie'=e'a_i=a_i$ for some $e^\prime 2=e' \in R$ . This implies that $a_iR$ is a nonzero submodule of $\mathrm {r.ann}_R(\mathrm {l.ann}_R(a_i))$ . Now we show that $a_iR$ is a unique simple submodule of $\mathrm {r.ann}_R(\mathrm {l.ann}_R(a_i))$ . It is enough to show that $a_iR \subseteq xR$ for any $0 \neq x \in \mathrm {r.ann}_R(\mathrm {l.ann}_R(a_i))$ . Assume that $0 \neq x \in \mathrm {r.ann}_R(\mathrm {l.ann}_R(a_i))$ . Then $\mathrm {l.ann}_R(a_i)x=0$ and so $\mathrm {l.ann}_R(a_i) \subseteq \mathrm {l.ann}_R(x)$ . Since R is a ring with local units, $x, a_i \in e"Re"$ for some ${e"}^2=e" \in R$ . From $e"x=x$ and $x \neq 0$ , we conclude that $\mathrm {l.ann}_R(x)$ is a proper submodule of R. By using the fact that $Ra_i \cong R/{\mathrm { l.ann}_R(a_i)}$ as R-modules and $Ra_i$ is a simple module, we can see that $\mathrm {l.ann}_R(a_i) = \mathrm {l.ann}_R(x)$ . Hence, we have an isomorphism $g: Rx \rightarrow Ra_i$ via $rx \mapsto ra_i$ . Since $Re_i$ is injective in RMod, there exists a morphism $h: R \rightarrow Re_i$ such that $(rx)h=(rx)g$ for each $r \in R$ . Set $y=(e")h$ . Then $y \neq 0$ since otherwise for any $r \in R$ , $(rx)g=(rx)h=(r(xe"))h=((rx)e")h=rx(e")h=rxy=0$ , which is a contradiction. Since $xy=x(e")h=(xe")h=(e"x)h=(e"x)g=e"a_i=a_i$ , we have $a_iR \subseteq xR$ . This shows that $a_iR$ is a unique simple submodule of $\mathrm { r.ann}_R(\mathrm {l.ann}_R(a_i))$ . Since $a_i \in Re_i$ , the mapping $\varphi : e_iR \rightarrow a_iR$ via $e_ir \mapsto a_ir$ is well-defined, and it is easy to see that it is an R-module epimorphism. Since $a_iR$ is simple and $e_i$ is a local idempotent, $\mathrm {Ker}\varphi =\mathrm {rad}(e_iR)$ and so $\mathrm {top}(e_iR) \cong a_iR$ as R-modules. Similarly, we can see that $\mathrm {Soc}(f_jR)=b_jR$ for some $b_j \in R$ , $Rb_j$ is the unique simple submodule of $\mathrm {l.ann}_R(\mathrm {r.ann}_R(b_j))$ and $\mathrm {top}(Rf_j) \cong Rb_j$ as R-modules. Now we show that for each $i,~(1 \leq i \leq n)$ , there exists $1 \leq j \leq m$ such that $\mathrm { Soc}(Re_i) \cong \mathrm {top}(Rf_j)$ and $\mathrm {Soc}(f_jR) \cong \mathrm {top}(e_iR)$ as R-modules. Since $fR$ is faithful, there is an R-module monomorphism $R \rightarrow \widehat {\prod }_A fR$ , where A is a set. This implies that there is an R-module homomorphism $\beta : R \rightarrow fR$ such that $\beta |_{a_iR} \neq 0$ . Since $a_iR$ is simple, $\beta |_{a_iR}: a_iR \rightarrow fR$ is a monomorphism. Since $fR$ is injective in ModR, there is an R-module monomorphism $E(a_iR) \rightarrow fR$ . This yields that $E(a_iR) \cong f_jR$ for some $1 \leq j \leq m$ . Therefore, $a_iR \cong \mathrm {Soc}(f_jR)$ as R-modules and so $\mathrm {top}(e_iR) \cong \mathrm { Soc}(f_jR)$ as R-modules. Now we show that $\mathrm {Soc}(Re_i) \cong \mathrm {top}(Rf_j)$ as R-modules. Since $a_iR \cong \mathrm {Soc}(f_jR)=b_jR$ as R-modules, $R\mathrm {Hom}_R(a_iR, R) \cong R\mathrm { Hom}_R(b_jR, R)$ as R-modules. It is not difficult to see that the mapping $R\mathrm {Hom}_R(a_iR, R) \rightarrow \mathrm {l.ann}_R(\mathrm {r.ann}_R(a_i))$ via $\alpha \mapsto \alpha (a_i)$ is an R-module isomorphism. Similarly, $R\mathrm {Hom}_R(b_jR, R) \cong \mathrm {l.ann}_R(\mathrm {r.ann}_R(b_j))$ as R-modules. Therefore, $\mathrm {l.ann}_R(\mathrm {r.ann}_R(b_j)) \cong \mathrm {l.ann}_R(\mathrm {r.ann}_R(a_i))$ as R-modules. This yields $Rb_j \cong \mathrm {Soc}(\mathrm {l.ann}_R(\mathrm {r.ann}_R(a_i)))$ as R-modules. Since $Ra_i \mathrm {r.ann}_R(a_i)=0$ , $Ra_i \subseteq \mathrm {l.ann}_R(\mathrm {r.ann}_R(a_i))$ . Since $Ra_i$ is simple, $\mathrm { Soc}(Re_i) \cong \mathrm {top}(Rf_j)$ as R-modules. Consequently, for each $i,~(1 \leq i \leq n)$ , there is $1 \leq j \leq m$ , such that $\mathrm {Soc}(Re_i) \cong \mathrm {top}(Rf_j)$ and $\mathrm {Soc}(f_jR) \cong \mathrm { top}(e_iR)$ as R-modules. Similarly, we can see that for each $l,~(1 \leq l \leq m)$ , there is $1 \leq k \leq n$ such that $\mathrm {Soc}(Re_k) \cong \mathrm {top}(Rf_l)$ and $\mathrm {Soc}(f_lR) \cong \mathrm {top}(e_kR)$ as R-modules. This shows that the result follows.

$(b)$ . Since $Re=\bigoplus _{i=1}^nRe_i$ , $fRe=\bigoplus _{i=1}^nfRe_i$ . By Lemma 4.1, $fRe$ is an injective left $fRf$ -module and hence each $fRe_i$ is an injective left $fRf$ -module. Also, by using Lemma 4.1, it is not difficult to see that $\mathrm {End}_R(Re_i) \cong \mathrm {End}_{fRf}(fRe_i)$ as rings. Since $\mathrm {End}_R(Re_i)$ is a local ring, $\mathrm {End}_{fRf}(fRe_i)$ is a local ring and so each $fRe_i$ is an indecomposable injective left $fRf$ -module. Now we show that $\mathrm {Soc}(fRe_i) \neq 0$ . By $(a)$ , $\mathrm {Soc}(Re_i) \cong \mathrm {top}(Rf_{\sigma (i)})$ as R-modules, where $\sigma $ is a permutation of $\lbrace 1, \cdots , n \rbrace $ . Hence, $f\mathrm {Soc}(Re_i) \cong f\mathrm {top}(Rf_{\sigma (i)})$ as $fRf$ -modules. If $f\mathrm {Soc}(Re_i)=0$ , then $f\mathrm {top}(Rf_{\sigma (i)})=0$ and so $f_{\sigma (i)} \in \mathrm {rad}(Rf_{\sigma (i)})$ . Hence, $\mathrm { Soc}(Re_i)=0$ , which is a contradiction. Thus, $f\mathrm {Soc}(Re_i)\neq 0$ and we can easily see that it is a simple $fRf$ -submodule of $fRe_i$ . Therefore, $\mathrm {Soc}(fRe_i) \neq 0$ . But because $fRe_i$ is an indecomposable injective left $fRf$ -module, $fRe_i$ has a simple essential socle. On the other hand, since $\mathrm {Soc}(fRe)=\bigoplus _{i=1}^n\mathrm {Soc}(fRe_i)$ , it is enough to show that $\lbrace \mathrm {Soc}(fRe_1), \cdots , \mathrm {Soc}(fRe_n)\rbrace $ is a complete set of nonisomorphic simple left $fRf$ -modules. Since $\lbrace fRf_1/\mathrm {rad}(fRf_1), \cdots , fRf_n/\mathrm {rad}(fRf_n)\rbrace $ is a complete set of nonisomorphic simple left $fRf$ -modules, it is sufficient to show that $\mathrm {Soc}(fRe_i) \cong fRf_{\sigma (i)}/\mathrm {rad}(fRf_{\sigma (i)})$ as $fRf$ -modules. It is not difficult to see that $f\mathrm {top}(Rf_{\sigma (i)}) \cong fRf_{\sigma (i)}/f\mathrm {rad}(Rf_{\sigma (i)})$ as $fRf$ -modules. But since $f\mathrm {top}(Rf_{\sigma (i)}) \cong f\mathrm {Soc}(Re_i)=\mathrm {Soc}(fRe_i)$ as $fRf$ -modules and $fRe_i$ has a simple socle, $\mathrm { rad}(fRf_{\sigma (i)}) = f\mathrm {rad}(Rf_{\sigma (i)})$ . This implies that $f\mathrm {top}(Rf_{\sigma (i)}) \cong \mathrm {top}(fRf_{\sigma (i)})$ as $fRf$ -modules. Therefore, $\mathrm {Soc}(fRe_i) \cong \mathrm {top}(fRf_{\sigma (i)})$ as $fRf$ -modules and the result follows.

Proof $(a)$ . Let M be a left $fRf$ -module and $Re$ , $e^2=e\in R$ , be a minimal faithful left ideal of R. By Proposition 4.2(b), $fRe$ is a cogenerator in $fRf$ -Mod. Then there exists an exact sequence in $fRf$ -Mod

$$ \begin{align*}0 \rightarrow M \rightarrow \prod_{I} {fRe} \rightarrow \prod_{J} {fRe},\end{align*} $$

where I and J are two index sets. By using Proposition 4.1, we can see that

$$ \begin{align*}R\mathrm{Hom}_{fRf}(fR,\prod_{I} fRe) \cong \widehat{\prod_{I}} Re,\end{align*} $$

as R-modules. This gives us the following exact sequence:

$$ \begin{align*}0 \rightarrow R\mathrm{Hom}_{fRf}(fR,M) \rightarrow \widehat{\prod_{I}}Re \rightarrow \widehat{\prod_{J}}Re,\end{align*} $$

Since R is a left perfect right locally coherent ring and $\mathrm {l.gl.dim}R =0 ~\mathrm {or}~ 2$ , $R\mathrm {Hom}_{fRf}(fR,M) \in \mathrm {Proj}(R)$ . Since R is a semiperfect ring with enough idempotents, there is a complete set $\lbrace e_{\alpha }~|~\alpha \in J \rbrace $ of pairwise orthogonal local idempotents of R such that $R=\bigoplus _{\alpha \in J}Re_{\alpha }=\bigoplus _{\alpha \in J}e_{\alpha }R$ . Hence, $R\mathrm {Hom}_{fRf}(fR,M) \cong \bigoplus _{\alpha \in A}Re_{\alpha }$ as R-modules. There are $fRf$ -module isomorphisms

$$ \begin{align*}M \cong fR \otimes_R R\mathrm{Hom}_{fRf}(fR,M) \cong fR \otimes_R (\bigoplus_{\alpha \in A}Re_{\alpha}) \cong \bigoplus_{\alpha \in A} (fR \otimes_R Re_{\alpha}) \cong \bigoplus_{\alpha \in A} fRe_{\alpha}.\end{align*} $$

Since $fR$ is balanced and faithful, the canonical ring homomorphism $\rho : R \rightarrow R\mathrm {End}_{fRf}(fR)$ via $r \mapsto [\rho _r: x \mapsto xr]$ is an isomorphism. Hence, $Re_{\alpha } \cong R\mathrm { End}_{fRf}(fR)e_{\alpha }$ as R-module. It is easy to see that the mapping $\varphi : R\mathrm {Hom}_{fRf}(fR, fRe_{\alpha }) \rightarrow R\mathrm {End}_{fRf}(fR)e_{\alpha }$ via $\alpha \mapsto \alpha \varepsilon e_{\alpha }$ is an isomorphism of R-modules, where $\varepsilon : fRe_{\alpha } \rightarrow fR$ is the canonical injection. It follows that $R\mathrm {Hom}_{fRf}(fR, fRe_{\alpha }) \cong Re_{\alpha }$ as R-modules and so $\mathrm { End}_R(Re_{\alpha }) \cong \mathrm {End}_{fRf}(fRe_{\alpha })$ as rings. Since $\mathrm {End}_R(Re_{\alpha })$ is a local ring, $\mathrm {End}_{fRf}(fRe_{\alpha })$ is a local ring and so each $fRe_{\alpha }$ is an indecomposable left $fRf$ -module. Therefore, every left $fRf$ -module is a direct sum of indecomposable modules. Consequently, by [Reference Krause32, Corollary 2.7], $fRf$ is a left pure semisimple ring.

$(b)$ . By using the proof of $(a)$ , we can see that functor $R\mathrm {Hom}_{fRf}(fR, -)$ sends $fRf$ -Mod to the full subcategory $\mathrm {Proj}(R)$ of RMod. The natural transformation

$$ \begin{align*} \eta: fR \otimes_R R\mathrm{Hom}_{fRf}(fR,-) \rightarrow \mathrm{id}\ \text{via}\ fr \otimes \alpha \mapsto (fr)\alpha, \end{align*} $$

for $r \in R$ and $\alpha \in R\mathrm {Hom}_{fRf}(fR,X)$ , where X is a left $fRf$ -module, is an equivalence. Therefore, it is enough to show that the natural transformation

$$ \begin{align*} \nu: \mathrm{id} \rightarrow R\mathrm{Hom}_{fRf}(fR, fR \otimes_R -)\ \text{via}\ x \mapsto [fr \mapsto fr \otimes x], \end{align*} $$

for $x \in X$ and $r \in R$ is an equivalence when X is a projective unitary left R-module. Let P be a projective unitary left R-module. Then $P \cong \bigoplus _{\alpha \in A}Re_{\alpha }$ as R-modules. We know that for each idempotent $e'$ of R, $fRe'$ is a finitely generated left $fRf$ -module, because $fRf$ is left pure semisimple. This implies that the functor $R\mathrm {Hom}_{fRf}(fR,-)$ preserves direct sums and so we have the following isomorphisms:

$$ \begin{align*} R\mathrm{Hom}_{fRf}(fR, fR \otimes_R P) & \cong R\mathrm{Hom}_{fRf}(fR, \bigoplus_{\alpha \in A} (fR \otimes_R Re_{\alpha})) \\ & \cong \bigoplus_{\alpha \in A} R\mathrm{Hom}_{fRf}(fR, fR \otimes_R Re_{\alpha}) \\ &\cong \bigoplus_{\alpha \in A} R\mathrm{Hom}_{fRf}(fR, fRe_{\alpha}) \cong \bigoplus_{\alpha \in A}Re_{\alpha} \cong P, \end{align*} $$

which imply that $\nu _{P}$ is an isomorphism. Therefore, the functor

$$ \begin{align*} R\mathrm{Hom}_{fRf}(fR,-): fRf\text{-Mod} \rightarrow \mathrm{Proj}(R), \end{align*} $$

is an additive equivalence with the inverse equivalence $fR \otimes _R-: \mathrm {Proj}(R) \rightarrow fRf$ -Mod.

Proof Since W is finitely generated, $W \in \mathrm {add}(U)$ and so U is a generator–cogenerator in Mod- $\Lambda $ . Hence, by using the proof of Proposition 3.3, we can see that $\widehat {\mathrm {Hom}}_{\Lambda }(U, W)$ is a minimal faithful unitary right R-module. Set $\Delta =\mathrm {End}_{\Lambda }(W)$ . Since $\widehat {\mathrm {Hom}}_{\Lambda }(U, W) \cong \bigoplus _{\alpha \in I}\mathrm {Hom}_{\Lambda }(U_{\alpha }, W)$ as $\Delta $ -modules and the functor $\mathrm {Hom}_{\Lambda }(-, W):$ mod- $\Lambda \rightarrow \Delta $ -mod is a duality, for each $\gamma \in R\mathrm {End}_{\Delta }(\widehat {\mathrm {Hom}}_{\Lambda }(U, W))$ , there exists $r \in R$ such that $\rho _r=\gamma $ . This implies that the canonical ring homomorphism $\rho : R \rightarrow R\mathrm {End}_{\Delta }(\widehat {\mathrm {Hom}}_{\Lambda }(U, W))$ via $r \mapsto [\psi \mapsto \psi r]$ is surjective.

Proof By Corollary 3.4, [Reference Fazelpour and Nasr-Isfahani15, Lemma 2.3] and [Reference Fuller17, Theorem], $R_{\Lambda }$ is a left perfect ring with enough idempotents such that $\mathrm {l.gl.dim}R_{\Lambda } = 0 ~\mathrm { or} ~ 2$ . Since W is Morita duality $\Lambda $ – $\Delta $ -bimodule (as discussed in [Reference Anderson and Fuller2, Section 24]), by [Reference Fuller and Hullinger18, Theorem 3.3] and [Reference Fazelpour and Nasr-Isfahani15, Lemma 2.3], $R_{\Lambda }$ is a QF-3 ring. Moreover, by [Reference Fuller and Hullinger18, Lemma 3.1] and [Reference Fazelpour and Nasr-Isfahani15, Lemma 2.3], $R_{\Lambda }$ is Morita equivalent to $R_{{\Delta }^{op}}$ . It is not difficult to see that the direct product of any family of projective unitary left $R_{\Lambda }$ -modules is projective. This implies that by [Reference Garcia and Simson23, Proposition 2.4], $R_{\Lambda }$ is a right locally coherent ring. Therefore, $R_{{\Delta }^{op}}$ is a left perfect QF-3 right locally coherent ring with enough idempotents with $\mathrm {l.gl.dim}R_{{\Delta }^{op}} =0 ~\mathrm {or}~ 2$ . On the other hand, since W is finitely generated minimal injective cogenerator in Mod- $\Delta $ and $\Delta $ is a right artinian ring, by Lemma 4.5, $R_{{\Delta }^{op}}$ has a minimal faithful balanced right ideal. Therefore, $R_{{\Delta }^{op}}$ is a left perfect right locally coherent QF-3 ring with enough idempotents, which have a minimal faithful balanced right ideal and $\mathrm {l.gl.dim}R_{{\Delta }^{op}} = 0 ~\mathrm {or} ~ 2$ .

Proof Let $\Lambda $ be a left pure semisimple ring. Then there exists a left pure semisimple basic ring $\Lambda '$ such that $\Lambda $ is Morita equivalent to $\Lambda '$ . Assume that W is a minimal injective cogenerator in $\Lambda '$ -Mod and $\Delta =\mathrm {End}_{\Lambda '}(W)$ . Then by Proposition 4.6, the right functor ring $R_{{\Delta }^{op}}$ of $\Delta $ is a left perfect right locally coherent QF-3 ring with enough idempotents, which have a minimal faithful balanced right ideal and $\mathrm {l.gl.dim}R_{{\Delta }^{op}} = 0 ~\mathrm {or} ~ 2$ . By using [Reference Fazelpour and Nasr-Isfahani16, Theorem 3.6], it is easy to see that the assignment $\Lambda \mapsto R_{{\Delta }^{op}}$ sends Morita equivalence classes of left pure semisimple rings to Morita equivalence classes of left perfect right locally coherent QF-3 rings $R_{{\Delta }^{op}}$ with enough idempotents, which have a minimal faithful balanced right ideal and $\mathrm {l.gl.dim}R_{{\Delta }^{op}} = 0 ~\mathrm {or} ~ 2$ . Also, this map is injective. Now we show that this map is surjective. Let R be a left perfect right locally coherent QF-3 ring with enough idempotents such that $\mathrm {l.gl.dim}R = 0 ~\mathrm {or} ~ 2$ . Assume that $fR$ , $f^2=f \in R$ , is a minimal faithful balanced right ideal of R. By Proposition 4.4, $fRf$ is a left pure semisimple ring and the functor $R\mathrm {Hom}_{fRf}(fR,-): fRf$ -Mod $\rightarrow \mathrm {Proj}(R)$ is an additive equivalence. Let Q be a minimal injective cogenerator in $fRf$ -Mod and $\Gamma =\mathrm {End}_{fRf}(Q)$ . It is not difficult to show that $R_{{\Gamma }^{op}}$ is Morita equivalent to R. Hence, this map is surjective and the result follows.

Proof By [Reference Simson40, Theorem 1.9], every finitely presented object in $\mathscr {A}$ is a finite coproduct of indecomposable subobjects and $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a perfect category and so $R_{\mathscr {A}}$ is a left perfect ring. Then there is a complete set $\lbrace \theta _i~|~i \in I \rbrace $ of pairwise orthogonal local idempotents of $R_{\mathscr {A}}$ such that $R_{\mathscr {A}}=\bigoplus _{i \in I}R_{\mathscr {A}}\theta _{i}=\bigoplus _{i \in I}\theta _{i}R_{\mathscr {A}}$ . Since $\mathbb {H}: \mathrm {fp}(\mathscr {A}) \rightarrow \mathrm {proj}(\mathrm {fp}(\mathscr {A}))$ is an additive equivalence, $\mathbb {G}: \mathrm { Mod}(\mathrm {fp}(\mathscr {A})) \rightarrow T_{\mathscr {A}}\mathrm {Mod}$ is an additive equivalence, which preserves and reflects finitely generated projective objects and $R_{\mathscr {A}}$ is Morita equivalent to $T_{\mathscr {A}}$ . By using [Reference Garcia and Simson24, Corollary 3], it is not difficult to show that each $\theta _{i}R_{\mathscr {A}}\theta _{i}$ is a left perfect ring and hence $R_{\mathscr {A}}$ has the property $(i)$ . Since $\mathscr {A}$ is a locally finitely presented pure semisimple Grothendieck category, by [Reference Simson39, Theorem 6.3], $\mathscr {A}$ is locally noetherian and every object in $\mathscr {A}$ is a coproduct of indecomposable noetherian objects. This implies that by [Reference Simson41, Corollary 2.6], $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is locally noetherian. By [Reference Stenström46, Proposition 4.3], $R_{\mathscr {A}}$ is left locally noetherian. On the other hand, since $\mathscr {A}$ is locally finitely presented Grothendieck category and $R_{\mathscr {A}}$ is a left perfect ring, by [Reference Garcia and Simson23, Theorems 2.7 and 3.3], $\mathrm {gl.dim}R_{\mathscr {A}}= 0~ \mathrm {or} ~2$ and $\mathrm {dom.dim}R_{\mathscr {A}}\theta _i \geq 2$ for each $i \in I$ . Therefore, $R_{\mathscr {A}}$ has the properties $(ii)$ and $(iv)$ . Now we show that $R_{\mathscr {A}}$ has the property $(iii)$ . Since $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, $R_{\mathscr {A}}$ is a left QF-3 ring and so there are finitely many pairwise nonisomorphic simple left ideals $S_1, \cdots , S_n$ such that $M = \bigoplus _{l=1}^n E(S_l)$ is faithful, projective, and injective in $R_{\mathscr {A}}$ Mod. We show that $\mathrm {proj}(R_{\mathscr {A}})\bigcap \mathrm {inj}(R_{\mathscr {A}})= \mathrm { add}(\bigoplus _{l=1}^n E(S_l))$ . Assume that $N \in \mathrm {proj}(R_{\mathscr {A}})\bigcap \mathrm {inj}(R_{\mathscr {A}})$ . Then $N \cong \bigoplus _{i=1}^tR_{\mathscr {A}}\theta _i$ . Since M is faithful, for each $i,~(1 \leq i \leq t)$ , there is a monomorphism $R_{\mathscr {A}}\theta _i \rightarrow \widehat {\prod }_{A} M$ , where A is a set. But since every representable functor has finitely generated essential socle, for each $i,~(1 \leq i \leq t)$ , there is a monomorphism $R_{\mathscr {A}}\theta _i \rightarrow \bigoplus _{l \in B_i} E(S_l)$ , where $B_i$ is a finite set. Since N is injective in $R_{\mathscr {A}}$ Mod and each $\mathrm {End}_{R_{\mathscr {A}}}(E(S_l))$ is a local ring, $N \in \mathrm {add}(\bigoplus _{l=1}^n E(S_l))$ and so $R_{\mathscr {A}}$ has the property $(iii)$ . Therefore, the result follows.

Proof We know that $\mathrm {Flat}(R)$ is a locally finitely presented category and $\mathrm {fp}(\mathrm {Flat}(R))$ coincides with the category of all finitely generated projective unitary left R-modules. By the proof of Theorem 3.6, we can see that there exists a left pure semisimple ring $\Lambda $ such that $R_{\Lambda }$ is Morita equivalent to R. It implies that by the proof of Proposition 4.6, R is a left perfect right locally coherent QF-3 ring. Properties $(ii)$ and $(iv)$ and [Reference Garcia and Simson23, Theorem 2.7] imply that $\mathrm {Flat}(R)$ is a Grothendieck category. Therefore, $\mathrm {Flat}(R)$ is a locally finitely presented Grothendieck category. Set $\mathscr {A}=\mathrm {Flat}(R)$ . Since $\mathrm {fp}(\mathscr {A})=\mathrm {proj}(R)$ , there is an additive equivalence $\mathrm {proj}(R) \rightarrow \mathrm {proj}(T_{\mathscr {A}})$ . By [Reference Fazelpour and Nasr-Isfahani16, Theorem 3.6], R is Morita equivalent to $T_{\mathscr {A}}$ . But because R is a left perfect left QF-3 ring and there is an additive equivalence $\mathrm {Mod}(\mathrm {fp}(\mathscr {A})) \rightarrow T_{\mathscr {A}}\mathrm {Mod}$ , $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a perfect QF-3 category. In particular, by [Reference Simson40, Theorem 1.9], $\mathscr {A}$ is a locally finitely presented pure semisimple Grothendieck category. Now we show that every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. Since R is a left locally noetherian ring with the properties $(i)$ and $(iii)$ , by using [Reference Stenström46, Example 18, p. 134], we can see that the injective hull of any finitely generated projective unitary left R-module is a finite direct sum of injective hull of simple modules. There is an additive equivalence $\mathrm {Mod}(\mathrm {fp}(\mathscr {A})) \rightarrow R\mathrm {Mod}$ , which preserves and reflects finitely generated projective objects and so by [Reference Stenström46, Example 25(ii), p. 134], every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. Therefore, $\mathrm {Flat}(R)$ is a locally finitely presented pure semisimple Grothendieck category, which satisfies the property  $(\ast )$ .

Proof Let R be a ring with the properties $(i)$ – $(iv)$ . By Proposition 5.2, $\mathrm {Flat}(R)$ is a locally finitely presented pure semisimple Grothendieck category, which satisfies the property $(\ast )$ . Let R and T be rings with the properties $(i)$ – $(iv)$ . If R and T are Morita equivalent, then there exists an additive equivalence $\mathrm {Flat}(R) \rightarrow \mathrm {Flat}(T)$ . It implies that the map $R \mapsto \mathrm {Flat}(R)$ sends Morita equivalence classes of rings with the properties $(i)$ – $(iv)$ to equivalence classes of locally finitely presented pure semisimple Grothendieck categories, which satisfy the property $(\ast )$ . We show that this mapping is a bijection. In order to show that this map is injective, we suppose that R and T are rings with the properties $(i)$ – $(iv)$ and there exists an additive equivalence $\mathrm {Flat}(R) \rightarrow \mathrm { Flat}(T)$ . Then there exists an additive equivalence $\mathrm {proj}(R) \rightarrow \mathrm {proj}(T)$ . By [Reference Fazelpour and Nasr-Isfahani16, Theorem 3.6], R is Morita equivalent to T and so this mapping is injective. It remains to show that this mapping is a surjective. Let $\mathscr {A}$ be a locally finitely presented pure semisimple Grothendieck category, which satisfies the property $(\ast )$ . Then by Proposition 5.1, $R_{\mathscr {A}}$ is a ring with the properties $(i)$ – $(iv)$ and also there is an additive equivalence $\mathscr {A} \rightarrow \mathrm {Flat}(R_{\mathscr {A}})$ . Therefore, the mapping is surjective and the result follows.

Proof By using [Reference Garcia and Simson24, Corollary 3], [Reference Garcia and Simson23, Theorem 2.7] and the existence of an additive equivalence $\mathscr {A} \rightarrow \mathrm {Flat}(R_{\mathscr {A}})$ , the functor ring $R_{\mathscr {A}}$ of $\mathscr {A}$ is a left artinian ring with $\mathrm {dom.dim}R_{\mathscr {A}} \geq 2$ and $\mathrm {l.gl.dim}R_{\mathscr {A}}= 0 ~\mathrm {or}~ 2$ and so $R_{\mathscr {A}}$ is a QF-3 ring. Since $R_{\mathscr {A}}$ is left artinian, every finitely generated projective left $R_{\mathscr {A}}$ -module has finitely generated essential socle. The existence of an additive equivalence $\mathrm {Mod}(\mathrm {fp}(\mathscr {A})) \rightarrow R_{\mathscr {A}}$ Mod, which preserves and reflects finitely generated projective objects, implies that $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category and every representable functor has finitely generated essential socle.

Proof Since R has the properties $(i)$ – $(iv)$ , by Proposition 5.2, $\mathscr {A}=\mathrm {Flat}(R)$ is a locally finitely presented pure semisimple Grothendieck category. It implies that by [Reference Simson39, Theorem 6.3], every indecomposable object in $\mathscr {A}$ is finitely generated. On the other hand, since R is a left artinian ring, $\mathscr {A}$ has only finitely many nonisomorphic indecomposable objects. In particular, by [Reference Dung and García12, Theorems 3.2 and 4.2], $\mathscr {A}$ is locally finite. Therefore, $\mathscr {A}$ is a locally finitely presented Grothendieck category of finite representation type.

Proof It follows from Corollary 3.7, Theorem 5.3, Lemmas 5.5 and 5.6, and the proof of Lemma 5.5.