The following results are useful in our statements, for the sake of the readability, we list them as lemmas.

#### Lemma 2.1

([12, Lemma 1]). *Let R be any ring*. *M is a R*-*module and* 0 → *M* → *I*_{0}(*M*) → *I*_{1}(*M*) → ⋯ *a minimal injective resolution for M*. *Then*, *for a simple R*-*module S and n* > 0,
$\begin{array}{}{\text{Ext}}_{R}^{n}(S,M)\cong {\text{Hom}}_{R}(S,{I}_{n}(M)).\end{array}$

#### Lemma 2.2

([12, Proposition 7]). *Let R be a Gorenstein ring with self-injective dimension n*. *Then for a simple left R*-*module S*, *if S is a submodule of I*_{n}(*R*), *S never appears in* soc(*I*_{i}(*R*)) *for all i* ≠ *n*.

#### Lemma 2.3

([8, Theorem 1.2]). *Let Λ be an n*-*Gorenstein noetherian algebra and* 0 ≤ l ≤ *n*−2. *Then* 𝓔_{l} *gives a duality between C*_{l}(*Λ*) *and C*_{l}(*Λ*^{op}), *where C*_{l}(*Λ*) *is the subcategory consisting of all module X* = 𝓔_{l}(*Y*) *for some Y* ∈ mod *Λ*^{op} with gr *Y* ≥ *l*, *and* 𝓔_{l} ∘ 𝓔_{l} *is isomorphic to the identity functor*.

#### Lemma 2.4

([8, Theorem 1.3]). *Let Λ be an n*-*Gorenstein noetherian algebra and* 0 ≤ *l* ≤ *n*. *Then* 𝓕_{l} *gives a duality between simple Λ*-*modules X with* gr *X* = *l and that of Λ*^{op}, *and* 𝓕_{l} ∘ 𝓕_{l} *is isomorphic to the identity functor*. *Moreover*, gr 𝓔_{l}(*X*)/𝓕_{l}(*X*) > *l holds*.

For any *A* ∈ mod *Λ*, there exists a projective presentation in mod *Λ*:
$$\begin{array}{}{P}_{1}\stackrel{f}{\to}{P}_{0}\to A\to 0.\end{array}$$

Then we get an exact sequence
$$\begin{array}{}0\to {A}^{\star}\to {P}_{0}^{\star}\stackrel{{f}^{\ast}}{\to}{P}_{1}^{\star}\to \text{Coker\hspace{0.17em}}{f}^{\star}\to 0\end{array}$$

in mod *Λ*^{op}. Recall from [13] that Coker *f*^{*} is called a *transpose* of *A*, and denoted by Tr *A*. We remark that the transpose of *A* depends on the choice of the projective presentation of *A*, but it is unique up to projective equivalence (see [13]). By [13, Proposition 2.6], we have the following two useful exact sequences:
$$\begin{array}{}0\to {\text{Ext}}_{{\mathit{\Lambda}}^{op}}^{1}(\text{Tr\hspace{0.17em}}A,\mathit{\Lambda})\to A\stackrel{{\sigma}_{A}}{\to}{A}^{\ast \ast}\to {\text{Ext}}_{{\mathit{\Lambda}}^{op}}^{2}(\text{Tr\hspace{0.17em}}A,\mathit{\Lambda})\to 0\end{array}$$(1)
$$\begin{array}{}0\to {\text{Ext}}_{\mathit{\Lambda}}^{1}(A,\mathit{\Lambda})\to \text{Tr\hspace{0.17em}}A\stackrel{{\sigma}_{\text{Tr\hspace{0.17em}}A}}{\u27f6}(\text{Tr\hspace{0.17em}}A{)}^{\ast \ast}\to {\text{Ext}}_{\mathit{\Lambda}}^{2}(A,\mathit{\Lambda})\to 0\end{array}$$(2)

The following result plays an important role in this paper.

#### Proposition 2.5

*Let Λ be an Auslander regular ring with* gl.dim *Λ* ≤ 3 *and S be a simple Λ*-*module with* gr *S* = 1. *Then* r. pd 𝓔_{1}(*S*) = gr 𝓔_{1}(*S*) = 1.

#### Proof

Let ⋯ → *P*_{1} → *P*_{0} → 𝓔_{1}(*S*) → 0 be a projective resolution of 𝓔_{1}(*S*) in mod *Λ*^{op} and *L* = Im(*P*_{1} → *P*_{0}). Then by [1, Theorem 3.7], gr 𝓔_{1}(*S*) ≥ 1 and we get an exact sequence
$\begin{array}{}0\to {P}_{0}^{\star}\to {L}^{\star}\to {\mathcal{E}}_{1}{\mathcal{E}}_{1}(S)\to 0\end{array}$ in mod *Λ*. Talking (, *Λ*), we have the following exact sequence in mod *Λ*^{op}:
$$\begin{array}{}0\to {L}^{\star \star}\to {P}_{0}^{\star \star}\to {\mathcal{E}}_{1}{\mathcal{E}}_{1}{\mathcal{E}}_{1}(S)\to {\mathcal{E}}_{1}({L}^{\star})\to 0.\end{array}$$

Put *H* = Coker (*L*^{**} →
$\begin{array}{}{P}_{0}^{\ast \ast}\end{array}$). We get the following commutative diagram with exact rows:
$$\begin{array}{}0\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}L\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}{P}_{0}\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{E}}_{1}(S)\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}0\phantom{\rule{thinmathspace}{0ex}},\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\downarrow \phantom{\rule{thinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}_{{\sigma}_{L}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\downarrow \phantom{\rule{thinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}_{{\sigma}_{{P}_{0}}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\downarrow \phantom{\rule{thinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}_{h}\\ 0\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}{L}^{\ast \ast}\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}{P}_{0}^{\ast \ast}\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{1em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}H\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}0\end{array}$$

where *h* is induced by the natural morphisms *σ L* and *σ P*_{0}. It is easy to see the morphism *h* is an epimorphism. Moreover, there is an exact sequence 0 → 𝓔_{2} (Tr *L*) → 𝓔_{1}(*S*) → *H* → 0 by snake lemma and the sequence (1).

Denote *S*_{r} = 𝓕_{1}(*S*), we claim that 𝓔_{2} (Tr *L*) = 0. By Lemma 2.4, 𝓕_{1} is a duality between simple *Λ*-module *S* with gr *S* = 1 and that of *Λ*^{op}. So we have gr *S*_{r} = 1. If 𝓔_{2} (Tr *L*) ≠ 0, *S*_{r} will be a submodule of 𝓔_{2} (Tr *L*) and hence gr *S*_{r} ≥ 2 by [1, Theorem 3.7] and [9, Lemma 2.3(2)], a contradiction will arise. So we have *L* ≅ *L*^{**} and *H* ≅ 𝓔_{1}(*S*).

By Lemma 2.3, we get 𝓔_{1}(*S*) ≅ 𝓔_{1}𝓔_{1}𝓔_{1}(*S*). This derives that 𝓔_{3} (Tr *L*) ≅ 𝓔_{1}(*L*^{*}) = 0 by the exactness of the sequence 0 → *H* → 𝓔_{1}𝓔_{1}𝓔_{1}(*S*) → 𝓔_{1}(*L*^{*}) → 0. Then the facts 𝓔_{i} (Tr *L*) = 0 for any *i* ≥ 1 and gl. dim *Λ* ≤ 3 yield that Tr *L* is projective. So we have *L* is projective and r.pd 𝓔_{1}(*S*) ≤ 1. It follows that r.pd 𝓔_{1}(*S*) = gr𝓔_{1}(*S*) = 1. □

We have the following useful result.

#### Corollary 2.6

*Let Λ be an Auslander regular ring with* gl. dim *Λ* ≤ 3 *and S be a simple Λ*-*module with* gr *S* = 1. *Then* 1. pd 𝓔_{1}𝓔_{1}(*S*) = gr 𝓔_{1}𝓔_{1}(*S*) = 1.

#### Proof

By Proposition 2.5, we get r.pd 𝓔_{1}(*S*) = gr 𝓔_{1}(*S*) = 1. Let 0 → *Q*_{1} → *Q*_{0} → 𝓔_{1}(*S*) → 0 be any projective resolution of 𝓔_{1}(*S*). Taking (, *Λ*), we have a projective resolution of 𝓔_{1}𝓔_{1}(*S*) in mod *Λ*: 0 →
$\begin{array}{}{Q}_{0}^{\ast}\to {Q}_{1}^{\ast}\end{array}$ 𝓔_{1}𝓔_{1}(*S*) → 0. Because gr 𝓔_{1}(*S*) = 1, we get that l.pd 𝓔_{1}𝓔_{1}(*S*) = 1. Moreover, we have gr 𝓔_{1}𝓔_{1}(*S*) ≥ 1 by [1, Theorem 3.7]. So we are done. □

Let *S* be any simple module with gr *S* = 1. Lemma 2.4 shows that the socle of 𝓔_{1}(*S*) is simple, denoted by *S*_{r}, and there is an exact sequence 0 → *S*_{r} → 𝓔_{1}(*S*) → *N* → 0 with gr *N* ≥ 2. On the other hand, we always have 0 → *S* → *S*^{**} → 𝓔_{2} (Tr *S*) → 0 for any *S* with gr *S* = 0. It is easy to see that gr 𝓔_{2} (Tr *S*) ≥ gr *S*+2. Then, it is natural to ask: does this hold true for any simple module with gr *S* = 1? The following result shows the relationship between *S* and 𝓔_{1}𝓔_{1}(*S*). That is, we have

#### Theorem 2.7

*Let Λ be an Auslander regular ring with* gl. dim *Λ* ≤ 3 *and S be a simple Λ*-*module with* gr *S* = 1. *Then there exists K* ∈ mod *Λ with* gr *K* ≥ 3 *such that* 0 → *S* → 𝓔_{1}𝓔_{1}(*S*) → *K* → 0 *is exact*.

#### Proof

Put *S*_{r} = 𝓕_{1}(*S*) and *N* = 𝓔_{1}(*S*_{r})/𝓕_{1}(*S*_{r}). By Lemma 2.4, we have gr *S*_{r} = 1 and gr *N* ≥ 2. Applying (−, *Λ*) on the exact sequence 0 → *S* $\stackrel{{i}_{S}}{\to}$ 𝓕_{1}(*S*_{r}) → *N* → 0, since gr *N* ≥ 2, we get the following exact sequence in mod *Λ*^{op}:
$$\begin{array}{}0\to {\mathcal{E}}_{1}{\mathcal{E}}_{1}({S}_{r})\to {\mathcal{E}}_{1}(S)\to {\mathcal{E}}_{2}(N)\to 0.\end{array}$$

Taking (−, *Λ*) again, we get an exact sequence 0 → 𝓔_{1}𝓔_{1}(*S*) → 𝓔_{1}𝓔_{1}𝓔_{1}(*S*_{r}) → 𝓔_{2}𝓔_{2}(*N*) → 0 in mod *Λ* by the fact gr 𝓔_{2}(*N*) ≥ 2. By Lemma 2.3, 𝓔_{1}(*S*) ≅ 𝓔_{1}𝓔_{1}𝓔_{1}(*S*). Suppose *f* and *g* are the isomorphisms 𝓔_{1}(*S*) → 𝓔_{1}𝓔_{1}𝓔_{1}(*S*) and 𝓔_{1}𝓔_{1}(*S*) → Ker(𝓔_{1}𝓔_{1}𝓔_{1}(*S*_{r}) → 𝓔_{2}𝓔_{2}(*N*)), respectively. Then we have the following commutative diagram with exact rows:
$$\begin{array}{}0\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}S\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{E}}_{1}({S}_{r})\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}N\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}}\phantom{\rule{thinmathspace}{0ex}}0\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\downarrow \phantom{\rule{thinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}_{u}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\downarrow \phantom{\rule{thinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}_{f}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\downarrow \phantom{\rule{thinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}_{v}\\ 0\u27f6{\mathcal{E}}_{1}{\mathcal{E}}_{1}(S)\u27f6{\mathcal{E}}_{1}{\mathcal{E}}_{1}{\mathcal{E}}_{1}({S}_{r})\u27f6{\mathcal{E}}_{2}{\mathcal{E}}_{2}(N)\u27f60,\end{array}$$

where *u* = *g*^{−1} ∘ *f* ∘ *i*_{S} and *v* is the induced morphism. Put *K* = Ker *v*, then we have exact sequences in mod *Λ*:
$$\begin{array}{}0\to S\to {\mathcal{E}}_{1}{\mathcal{E}}_{1}(S)\to K\to 0\phantom{\rule{1em}{0ex}}(\ast )\\ 0\to \mathit{\Lambda}\to N\to {\mathcal{E}}_{2}{\mathcal{E}}_{2}(N)\to 0\phantom{\rule{1em}{0ex}}(\ast \ast )\end{array}$$

by snake lemma. By [9, Lemma 2.3(2)] and the exactness of sequence (**), we have gr *K* ≥ 2. Applying (−, *Λ*) on the exact sequence (*), we get the following long exact sequence in mod *Λ*^{op}:
$$\begin{array}{}0\to {\mathcal{E}}_{1}(K)\to {\mathcal{E}}_{1}{\mathcal{E}}_{1}{\mathcal{E}}_{1}(S)\to {\mathcal{E}}_{1}(S)\to {\mathcal{E}}_{2}(K)\to {\mathcal{E}}_{2}{\mathcal{E}}_{1}{\mathcal{E}}_{1}(S)\to \cdots .\end{array}$$

By Corollary 2.6, we have l.pd 𝓔_{1}𝓔_{1}(*S*) = 1 and, therefore, 𝓔_{2}𝓔_{1}𝓔_{1}(*S*) = 0. So the facts 𝓔_{1}(*K*) = 0 and 𝓔_{1}𝓔_{1}𝓔_{1}(*S*)≅ 𝓔_{1}(*S*) yield that 𝓔_{2}(*K*) = 0. That is, gr *K* ≥ 3.□

Now we fix our sight on diagonal rings. We have the following

#### Theorem 2.8

*Let Λ be an Auslander regular ring. Then the following statements are equivalent*.

*Λ is diagonal*.

*Any simple Λ*-*module S with* gr *S* = *k satisfies* 1. pd *S* = *k for any k* ≥ 0.

*Any simple Λ*-*module S with* 1. pd *S* = *k satisfies S*≅ 𝓔, 𝓔, (*S*) *for any k* ≥ 0.

#### Proof

(1) ⇔ (2) is trivial by the definition of diagonal ring and Lemma 2.1. We need to show (1) ⇔ (3).

(1) ⇒ (3) Suppose *Λ* is diagonal. Let *S* be any simple *Λ*-module with l.pd *S* = *k*. Then one gets that 𝓔_{k}(*S*) ≠ 0. By Lemma 2.1, *S* can be embedded into *I*_{k}(*Λ*). So, *S* will never appear in *I*_{i}(*Λ*) for any *i* ≠ *k*. Applying Lemma 2.1 again, we get gr *S* = *k*. That is, *S*≅ 𝓔_{k}𝓔_{k}(*S*).

(3) ⇒ (1) Suppose any simple *Λ*-module *S* with 1. pd *S* = *k* satisfies *S* ≅ 𝓔_{k}𝓔_{k}(*S*) for any *k* ≥ 0. Then by [1, Theorem 3.7], we get gr *S* ≥ *k*. Hence *S* can not be embedded into any term of the minimal injective resolution except for *I*_{k}(*Λ*). That is to say, *Λ* is diagonal.□

Actually, we have a strong version of the above result.

#### Proposition 2.9

*Let Λ be an Auslander regular ring with global dimension n. Then, Λ is diagonal if and only if any simple Λ-module S with* gr *S* = *k satisfies* 1. pd *S* = *k for any k* ≤ *n*−2.

#### Proof

We need only to show the necessity. It is known that any simple module *S* with gr *S* = *n* must have a projective dimension *n*. Let *S* be any simple module with gr *S* = *n*−1. Then the projective dimension of *S* must not be *n* by Lemma 2.2. So we have l.pd *S* = *n*−1. That is, *Λ* is diagonal by Proposition 2.5 (2).□

Let *Λ* be an Auslander regular ring with global dimension at most 3. By Corollary 2.6, we get that any simple *Λ*-module *S* with gr *S* = 1 satisfies l.pd (𝓔_{1}𝓔_{1}(*S*)) = 1. So we have the following

#### Corollary 2.10

*Let Λ be an Auslander regular ring with global dimension at most* 3. *Then the following statements are equivalent*.

*Λ is diagonal*.

*Any simple Λ*-*module S with* gr *S* = 0 *is projective and any simple Λ*-*module S with* gr *S* = 1 *satisfies S* ≅ 𝓔_{1}𝓔_{1}(*S*).

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