NAKAYAMA ISOMORPHISMS FOR THE MAXIMAL QUOTIENT RING OF A LEFT HARADA RING

Kazuaki Nonomura

Abstract. From several results of Kado and Oshiro, we see that if the maximal quotient ring of a given left Harada ring R of type (∗) has a Nakayama automorphism, then R has a Nakayama isomorphism. This result poses a question whether if the maximal quotient ring of a given left Harada ring R has a Nakayama isomorphism, then R has a Nakayama isomorphism. In this paper, we shall show that a basic ring of the maximal quotient ring of a given Harada ring has a Nakayama isomorphism if and only if its Harada ring has a Nakayama isomorphism.

Introduction

Let R be a basic left Harada ring. Then we have a complete set

{e_{11}, …, e_{1n(1)}, …, e_{m1}, …, e_{mn(m)}}

of primitive idempotents for R such that for each i = 1, …, m (a) e_{i1}R is injective as a right R-module;

(b) J(e_{i,k−1}R) ≅ e_{ik}R for each k = 2, …, n(i).

We call R a ring of type (∗) if there exists an unique g_{i} in {e_{in(i)}}_{i=1}^{m} for each i = 1, …, m such that the socle of e_{i1}R is isomorphic to g_{i}R/J(g_{i}R) and the socle of Rg_{i} is isomorphic to Re_{i1}/J(Re_{i1}).

Oshiro [10] showed the following;

Result A ([10, Theorem 2]). Suppose that R is a left Harada ring which is not of type (∗). Then there exists a series of left Harada rings and surjective ring homomorphisms:

T_{1} ↠^{ϕ_{1}} T_{2} ↠^{ϕ_{2}} ⋯ ↠^{ϕ_{n−1}} T_{n} ↠^{ϕ_{n}} R

such that (1) T_{1} is of type (∗), and

(2) Ker ϕ_{i} is a simple ideal of T_{i} for any i ∈ {1, …, n}.

Kado and Oshiro [7] showed the following results;

Result B ([7, Proposition 5.3]). If every basic QF rings has a Nakayama automorphism, then every basic left Harada ring of type (∗) has a Nakayama isomorphism.

The detailed version of this paper will be submitted for publication elsewhere.

Result C ([7, Proposition 5.4]). Let S be a two-sided ideal of R that is simple as a left ideal and as a right ideal. If R has a Nakayama isomorphism, then R/S has a Nakayama isomorphism.

Moreover Kado showed the following;

Result D ([6, Corollary]). The maximal quotient ring of a left Harada ring of type (∗) is a QF ring.

Using these four results, we see that if the maximal quotient ring of a given left Harada ring R of type (∗) has a Nakayama automorphism, then R has a Nakayama isomorphism. So this result poses a question whether if the maximal quotient ring of a given left Harada ring R has a Nakayama isomorphism, then R has a Nakayama isomorphism. In this paper, we shall show that the maximal quotient ring of a given left Harada ring R has a Nakayama isomorphism iff R has a Nakayama isomorphism.

Throughout this paper, we assume that all rings are associative rings with identity and all modules are unitary. By M_{R} (resp. _{R}M), we means that M is a right (resp. left) R-module, respectively. We denote the set of primitive idempotents of R by Pi(R), and denote a complete set of primitive idempotents of R by pi(R).

We call a one-sided artinian ring R right (resp. left) QF-3 ring if E(R_{R}) (resp. E(_{R}R)) is projective, respectively.

We denote the maximal left (resp. right) quotient ring of R by Q_{ℓ}(R) (resp. Q_{r}(R)), respectively, and denote the maximal left and maximal right quotient ring of R by Q(R). If a ring is QF-3, its maximal left quotient ring and its right quotient ring coincide by [16, Theorem 1.4].

1. Maximal quotient ring

We list some basic results, which several authors showed, for our main result in this paper. Recall that for e, f ∈ Pi(R), we say that the pair (eR : Rf) is an i-pair if S(eR) ≅ fR/J(fR) and S(Rf) ≅ Re/J(Re).

Lemma 1 ([5]). Let R be a one-sided artinian ring, and let e ∈ Pi(R). Then the following conditions are equivalent:

(1) eR is injective.

(2) There exists some f ∈ Pi(R) such that (eR : Rf) is an i-pair. In this case, Rf is also injective.

Let R be a left perfect ring. Then R has a primitive idempotent e with S(R_{R})e ≠ 0. If R is QF-3, then the primitive idempotent e with S(R_{R})e ≠ 0 are characterized as follows;

Lemma 2 ([4, Theorem 2.1]). Let R be a one-sided artinian QF- 3 ring, and let e ∈ Pi(R). Then _{R}Re is injective if and only if S(R_{R})e ≠ 0.

We call e ∈ Pi(R) right (resp. left) S-primitive if S(R_{R})e ≠ 0 (resp. e S(_{R}R) ≠ 0), respectively.

The following statement, which Storrer [15, Proposition 4.8] showed, is helpful in this paper.

Lemma 3 ([15, Proposition 4.8]). Let R and Q = Q(R) be left perfect. Then (1) If e is a right S-primitive idempotent for R, then so is it for Q.

(2) if e_{1}, e_{2} are right S-primitive idempotents for R, then e_{1}R ≅ e_{2}R if and only if e_{1}Q ≅ e_{2}Q.

(3) If e is a right S-primitive idempotent for Q, then there exists a right S-primitive idempotent e′ ∈ R such that eQ ≅ e′Q.

A ring R is called a left Harada ring if it is left artinian and its complete set pi(R) of orthogonal primitive idempotents is arranged as follows:

pi(R) = ⋃_{i=1}^{m}{e_{ij}}_{j=1}^{n(i)},

where (a) each e_{i1}R_{R} is an injective module for each i = 1, 2, …, m.

(b) e_{i,k−1}R_{R} ≅ e_{ik}R, or J(e_{i,k−1}R_{R}) ≅ e_{ik}R for each i and each k = 2, 3, …, n(i).

(c) e_{ik}R ≅ e̸_{jt}R for i ≠ j.

Remark 1. Let R be a left Harada ring. Then Q(R) is also a left Harada ring (See [6, Theorem 4]) and a complete set pi(Q) of orthogonal primitive idempotents for Q coincides with pi(R) (See [6, p.248]).

Using Remark 1, Kado showed the following;

Proposition 4 ([6, Proposition 2]). Let R be a left Harada ring, and let (eR : Rf) be an i-pair for e, f ∈ pi(R). Then (eQ(R) : Q(R)f) is an i-pair .

Recall the following notation [6, p.249]. Let θ : fR → eR be an R-monomorphism such that Im θ = J(eR), where e, f ∈ Pi(R). Then by [15, Proposition 4.3], θ can be uniquely extended to a Q_{r}(R)-homomorphism θ^{∗} : fQ_{r}(R) → eQ_{r}(R).

We shall need the following results.

Lemma 5 ([6, Proposition 3]). Let R be a basic and left Harada ring, and Q = Q(R) and θ as above. Then the following hold.

(1) If e is not right S-primitive, then the extension θ^{∗} : fQ → eQ is an isomorphism.

(2) If e is right S-primitive, then the extension θ^{∗} : fQ → eQ is a monomorphism such that Im θ^{∗} = J(eQ).

Remark 2 (cf. [15, Lemma 4.2]). Let {g_{i}} ∪ {f_{j}} be a complete set of orthogonal primitive idempotents for R, where the g_{i} are right S-primitive and the f_{j} are not right S-primitive. We denote g_{0} by g_{0} = ∑ g_{i}. Then Q(R)g_{0} = Rg_{0} and Q(R)g = Rg for every right S-primitive idempotent g of R.

Let R be a basic left artinian ring, and let {e_{1}, e_{2}, …, e_{n}} be a complete set of orthogonal primitive idempotents for R and let

S = End_{R}(⊕_{i=1}^{n}E(Re_{i}/J(Re_{i})))

be the endomorphism ring of a minimal injective cogenerator for R-mod. Let f_{i} be the primitive idempotent for S corresponding to the projection

⊕_{i=1}^{n}E(Re_{i}/J(Re_{i})) → E(Re_{i}/J(Re_{i})).

Then we call a ring isomorphism τ : R → S a Nakayama isomorphism if τ(e_{i}) = f_{i} for each i = 1, 2, …, n. By [3, p.42], the existence of a Nakayama isomorphism does not depend on the choice of the complete set {e_{1}, e_{2}, …, e_{n}} of orthogonal primitive idempotents. (See [7, Remark on p.387].)

It is important whether the maximal quotient ring of a basic artinian ring is basic since a Nakayama isomorphism is defined on a basic ring. Here we shall study the case that the maximal quotient ring of a given left Harada ring is basic.

Theorem 6 (cf. [2, Corollary 22]). Let R be a basic and left Harada ring and Q = Q(R). Then Q is a basic ring if and only if R either is QF or satisfies the following; n(i) = 1 or 2 and Re_{i1} is injective for any i. In this case R = Q.

Proof. Note that both R and Q are artinian QF-3. Let pi(R) = ⋃_{i=1}^{m}{e_{ij}}_{j=1}^{n(i)} be a complete set of orthogonal primitive idempotent for R satisfying the following conditions:

(a) e_{i1}R_{R} is injective for each i = 1, 2, …, m,

(b) e_{i,j+1}R_{R} ≅ J(e_{i}R_{R}) for j = 1, 2, …, n(i) − 1.

We have a complete set {Rg_{1}, …, Rg_{m}} of pairwise non-isomorphic indecomposable injective projective left R-modules, such that the (e_{i1}R : Rg_{i}) are i-pair for each i = 1, …, m since R is basic and artinian QF-3.

Assume that Q is basic. Let e_{i,k+1}, e_{ik} ∈ {e_{ij}}_{j=2}^{n(i)}. Then we have an R-monomorphism θ_{ik} : e_{i,k+1}R → e_{ik}R such that Im θ = J(e_{ik}R). If e_{ik} is not right S-primitive, then e_{ik+1}Q ≅ e_{ik}Q by Lemma 5. This contradicts that Q is basic. Hence e_{ik} is right S-primitive for k = 1, 2, …, n(i) − 1. Since the Re_{ik} are injective for each k = 1, 2, …, n(i) − 1 by Lemma 2, there exists some Rg in {Rg_{1}, …, Rg_{m}} such that Re_{ik} ≅ Rg. However R is basic, so we see that n(i) ≤ 2 and e_{i1} is right S-primitive.

In case n(i) = 1 for every i = 1, …, m, then R is QF.

In case n(i) = 2 for some i ∈ {1, …, m}. If e_{in(i)} is right S-primitive, then _{R}Re_{in(i)} is injective by Lemma 2. Hence e_{in(i)} is not right S-primitive since _{R}Re_{i1} is injective and so {Rg_{1}, …, Rg_{m}} = {Re_{11}, …, Re_{m1}}.

Conversely, first, assume that R is QF. Since _{R}Re is injective for any e ∈ pi(R), e is right S-primitive by Lemma 2. Thus, eQ ≅ f̸Q for any e, f ∈ pi(R) = pi(Q) by Lemma 3. Therefore Q is basic. Next, assume that R satisfies n(i) = 1 or 2 and Re_{i1} is injective for any i. Then e_{i1} is left S-primitive and so eQ = eR by Remark 2. Hence J(eQ) = J(eR). Therefore it is also clear to see that R = Q. □

Example 1. We shall give a basic left Harada ring R with J(R)^{5} = 0, which is not QF. Let R be an algebra over a field K defined by the following quiver;

[figure: 1; 2; 3; 4; α; β; γ; β′; γ′]

with the relations γβ = γ′β′, αγβ = 0, and β′αγ = 0.

The composition diagrams of the Loewy factors of the indecomposable projective modules of R_{R} is the following.

[figure: eR/eJ; eJ/eJ^{2}; eJ^{2}/eJ^{3}; eJ^{4}; 1; 2; 3; 4; 1; 2; 3; 4; 1; 3; 1; 2; 4; 4; 1; 2; 3]

Then R is a left Harada ring which is not QF since e_{1}R_{R}, e_{3}R_{R} and e_{4}R_{R} are injective and e_{2}R_{R} ≅ J(e_{1}R). Moreover e_{1}, e_{3}, e_{4} are right S-primitive. Hence e_{1}Q(R) = e_{1}R, e_{3}Q(R) = e_{3}R and e_{4}Q(R) = e_{4}R are injective and e_{2}Q(R) ≅ J(e_{1}Q(R)). Therefore R = Q(R).

Example 2. We shall give a basic Harada ring R with J(R)^{6} = 0, but Q(R) is not basic. Let R be an algebra over a field K defined by the following quiver;

[figure: 1; 2; 3; 4; α; β; γ; β′; γ′]

with the relations 0 = βαγβ = β′αγ′β′ = βαγ = β′αγ′, and γβ = γ′β′. Then the composition diagrams of the Loewy factors of the indecomposable projective modules of R_{R} is the following.

[figure: e_{i}R/e_{i}J; e_{i}J/e_{i}J^{2}; e_{i}J^{2}/e_{i}J^{3}; e_{i}J^{4}/e_{i}J^{5}; e_{i}J^{5}; 1; 2; 3; 4; 1; 2; 2; 3; 4; 1; 2; 3; 1; 2; 4; 4; 1; 2; 3]

Then since e_{1}R_{R}, e_{3}R_{R} and e_{4}R_{R} are injective and e_{2}R_{R} ≅ J(e_{1}R), R is a left Harada ring which is not QF. Hence e_{2}Q(R) ≅ e_{1}Q(R) since e_{1} is not right S-primitive. Therefore Q(R) is not basic.

2. Nakayama isomorphism

In this section, we study the Nakayama isomorphisms for the representative matrix ring of a basic left Harada ring and its maximal quotient ring. Let R be a basic left Harada ring, and let pi(R) = ⋃_{i=1}^{m}{e_{ij}}_{j=1}^{n(i)} be a complete set of orthogonal primitive idempotents as in Theorem 6. Furthermore, let R^{∗} be the representative matrix ring of R. R^{∗} is represented as block matrices as follows:

R^{∗} = (\begin{matrix}R_{11}^{∗} & ⋯ & R_{1m}^{∗} \\ & ⋯ \\ R_{m1}^{∗} & ⋯ & R_{mm}^{∗} \end{matrix}) ,

where R_{ij}^{∗} = P_{ij} for j ≠ σ(i) and R_{iσ(i)}^{∗} = P_{iσ(i)}^{∗} (See [7, Section 4]).

Here, adding one row and one column to R^{∗}, we make an extended matrix ring W_{i}(R) of R as follows:

(\begin{matrix}R_{11}^{∗} & ⋯ & … & R_{1i}^{∗} & Y_{1} & R_{1,i+1}^{∗} & ⋯ & R_{1m}^{∗} \\ ⋮ & & & ⋮ & ⋮ & ⋮ & & ⋮ \\ R_{i1}^{∗} & ⋯ & ⋯ & R_{ii}^{∗} & Y_{i} & R_{i,i+1}^{∗} & ⋯ & R_{im}^{∗} \\ X_{1} & ⋯ & X_{i−1} & X_{i} & Q & X_{i+1} & ⋯ & X_{m} \\ R_{i+1,1}^{∗} & ⋯ & ⋯ & R_{i+1,i}^{∗} & Y_{i+1} & R_{i+1,i+1}^{∗} & ⋯ & R_{i+1,m}^{∗} \\ ⋮ & & & ⋮ & ⋮ & ⋮ & & ⋮ \\ R_{m1}^{∗} & ⋯ & ⋯ & R_{mi}^{∗} & Y_{m} & R_{m,i+1}^{∗} & ⋯ & R_{mm}^{∗} \end{matrix}) ,

where X_{k} is the last row of R_{ik}^{∗} (k = 1, …, m, k ≠ i), Y_{k} is the last column of R_{ki}^{∗} (k = 1, …, m), X_{i} = (P_{in(i),i1}^{∗} … P_{in(i),in(i)−1}^{∗}J(P_{in(i),in(i)}^{∗})), and Q = P_{in(i),in(i)}^{∗}.

Then W_{i}(R) naturally becomes a ring by operations of R^{∗}. We call this the i-th extended ring of R.

Proposition 7 ([7, Proposition 5.11]). If W_{i}(R) has a Nakayama isomorphism, then R also has a Nakayama isomorphism.

Let R be a basic and left Harada ring, and let

pi(R) = ⋃_{i=1}^{m}{e_{ij}}_{j=1}^{n(i)}

be a complete set of orthogonal primitive idempotents of R satisfying the following; (1) e_{i1}R_{R} is injective for each i = 1, 2, …, m.

(2) e_{ij}R ≅ J(e_{i,j−1}R) for each j = 2, …, n(i). Then (See [7, p.388]), for any e_{ij} in pi(R), there exists some g_{i} in pi(R) with Rg_{i} injective such that E(Re_{ij}/J(Re_{ij})) ≅ Rg_{i}/S_{j−1}(Rg_{i}), where S_{j}(Rg_{i}) is the j-th socle of Rg_{i}. We denote the generator g_{i} + S_{j−1}(Rg_{i}) of Rg_{i}/S_{j−1}(Rg_{i}) by g_{ij} for each i = 1, 2, …, m, j = 1, 2, …, n(i). Then by [7, Proposition 3.2], a minimal injective cogenerator G = ⊕_{i,j}Rg_{ij} is finitely generated. Therefore we note that R is left Morita dual to End_{R}(G) by [1, Theorem 30.4]. We call this End(_{R}G) the dual ring of R. We denote the dual ring of R by T (R).

For the proof of proposition 8 below, we denote

(\begin{matrix}0 & & ⋯ & & 0 \\ 0 & ⋯ 0 & R_{ij}^{∗} & 0 ⋯ & 0 \\ 0 & & ⋯ & & 0 \end{matrix}) ⊆ R^{∗}

by [R_{ij}^{∗}] and

(\begin{matrix}0 & & ⋯ & & 0 \\ 0 & ⋯ 0 & R_{ij}^{∗} & 0 ⋯ & 0 \\ 0 & & ⋯ & & 0 \end{matrix}) ⊆ W_{i}(R)

by [R_{ij}^{∗}]^{w}.

By using the result that Kado and Oshiro [7, Proposition 5,11] showed, we shall show the following proposition. The proposition is essential in this paper.

Proposition 8. W_{i}(R) has a Nakayama isomorphism if and only if so does R.

Proof. (⇒). By Proposition 7 ([7, Proposition 5,11]). (⇐). As [7, Proposition 5.11], let e_{ij} be the matrix of R^{∗} such that the (ij, ij)-component is the unity and other components are zero, and let w_{ij} be the matrix of W_{i}(R) such that the (ij, ij)-component is the unity and other components are zero. Note that the size of the columns in W_{i}(R) is n(i) + 1. Let Ψ be the natural embedding homomorphism;

(\begin{matrix}R_{11}^{∗} & ⋯ & R_{1m}^{∗} \\ & ⋯ \\ R_{m1}^{∗} & ⋯ & R_{mm}^{∗} \end{matrix})

↓ Ψ

\overbrace{i + 1}^{(\begin{matrix}R_{11}^{∗} & ⋯ & … & R_{1i}^{∗} & 0 & R_{1,i+1}^{∗} & ⋯ & R_{1m}^{∗} \\ ⋮ & & & ⋮ & ⋮ & ⋮ & & ⋮ \\ R_{i1}^{∗} & ⋯ & ⋯ & R_{ii}^{∗} & 0 & R_{i,i+1}^{∗} & ⋯ & R_{im}^{∗} \\ 0 & ⋯ & 0 & 0 & 0 & 0 & ⋯ & 0 \\ R_{i+1,1}^{∗} & ⋯ & ⋯ & R_{i+1,i}^{∗} & 0 & R_{i+1,i+1}^{∗} & ⋯ & R_{i+1,m}^{∗} \\ ⋮ & & & ⋮ & ⋮ & ⋮ & & ⋮ \\ R_{m1}^{∗} & ⋯ & ⋯ & R_{mi}^{∗} & 0 & R_{m,i+1}^{∗} & ⋯ & R_{mm}^{∗} \end{matrix}) ,}

where R_{ij}^{∗} → R_{ij}^{∗} are identity maps for all i, j. Moreover let h_{ij} be the matrix of T (R) such that the (ij, ij)-component is the unity and other components are zero, and let v_{ij} be the matrix of W_{i}(T (R)) such that the (ij, ij)-component is the unity and other components are zero. Note that the size of the columns in W_{i}(T (R)) is n(i) + 1. Let

(\begin{matrix}T (R)_{11} & ⋯ & T (R)_{1m} \\ & ⋯ \\ T (R)_{m1} & ⋯ & T (R)_{mm} \end{matrix})

be the representative matrix ring T (R)^{∗} of T (R), and let T (W_{i}(R)) be the dual ring of W_{i}(R) as follows;

(\begin{matrix}T (R)_{11} & ⋯ & T (R)_{1i} & ^{t}Y_{1} & T (R)_{1,i+1} & ⋯ & T (R)_{1m} \\ ⋮ & & ⋮ & ⋮ & ⋮ & & ⋮ \\ T (R)_{i1} & ⋯ & T (R)_{ii} & ^{t}Y_{i} & T (R)_{i,i+1} & ⋯ & T (R)_{im} \\ ^{t}X_{1} & ⋯ & ^{t}X_{i} & ^{t}Q & ^{t}X_{i+1} & ⋯ & ^{t}X_{m} \\ T (R)_{i+1,1} & ⋯ & T (R)_{i+1,i} & ^{t}Y_{i+1} & T (R)_{i+1,i+1} & ⋯ & T (R)_{i+1,m} \\ ⋮ & & ⋮ & ⋮ & ⋮ & & ⋮ \\ T (R)_{m1} & ⋯ & T (R)_{mi} & ^{t}Y_{m} & T (R)_{m,i+1} & ⋯ & T (R)_{mm} \end{matrix}) .

Letting Ψ_{T (R)} be the natural embedding homomorphism;

(\begin{matrix}T (R)_{11} & ⋯ & T (R)_{1m} \\ & ⋯ \\ T (R)_{m1} & ⋯ & T (R)_{mm} \end{matrix})

↓ Ψ_{T (R)}

\overbrace{i + 1}^{(\begin{matrix}T (R)_{11} & ⋯ & … & T (R)_{1i} & 0 & T (R)_{1,i+1} & ⋯ & T (R)_{1m} \\ ⋮ & & & ⋮ & ⋮ & ⋮ & & ⋮ \\ T (R)_{i1} & ⋯ & ⋯ & T (R)_{ii} & 0 & T (R)_{i,i+1} & ⋯ & T (R)_{im} \\ 0 & ⋯ & 0 & 0 & 0 & 0 & ⋯ & 0 \\ T (R)_{i+1,1} & ⋯ & ⋯ & T (R)_{i+1,i} & 0 & T (R)_{i+1,i+1} & ⋯ & T (R)_{i+1,m} \\ ⋮ & & & ⋮ & ⋮ & ⋮ & & ⋮ \\ T (R)_{m1} & ⋯ & ⋯ & T (R)_{mi} & 0 & T (R)_{m,i+1} & ⋯ & T (R)_{mm} \end{matrix}) ,}

where T (R)_{ij} → T (R)_{ij} are identity maps for all i, j. We note that T (W_{i}(R)) = W_{i}(T (R)) (See [7, Proposition 5.11]).

Assume that φ : R^{∗} → T (R)^{∗} is a Nakayama isomorphism with φ(e_{ij}) = h_{ij}. (i.e., φ([r_{kl}]) ∈ [T (R)_{kl}] for any [r_{kl}] ∈ [R_{ij}^{∗}], where (k, l)-componentwise of R_{ij}^{∗} corresponds to (k, l)-componentwise of T (R)_{ij}.) We consider the following diagram;

W_{i}(R)  W_{i}(T (R))
Ψ↑  ↑^{Ψ_{T (R)}}
R^{∗} ⟶^{φ} T (R).

Here we define a map φ̄ : W_{i}(R) → W_{i}(T (R)) as follows;

(a) φ̄([r_{kl}]^{w}) = [φ([r_{kl}])]^{w} ∈ [T (R)_{kl}]^{w} for any [r_{kl}]^{w} ∈ [R_{kl}^{∗}]^{w}; 1 ≤ k ≤ m, 1 ≤ l ≤ m;

(b) φ̄([x]^{w}) ∈ [^{t}X_{k}]^{w} for any [x]^{w} ∈ [X_{k}]; k = 1, …, m;

(c) φ̄([y]^{w}) ∈ [^{t}Y_{l}]^{w} for any [y]^{w} ∈ [Y_{l}]^{w}; l = 1, …, m;

(d) φ̄([q]^{w}) ∈ [^{t}Q]^{w} for any [q]^{w} ∈ [Q]^{w}.

Since φ(e_{ij}) = h_{ij}, φ̄ is well-defined. Moreover it is satisfied φ̄(w_{i,n(i)+1}) = f_{i,n(i)+1}. Then we can easily check that φ̄ is a Nakayama isomorphism. □

Remark 3. We shall define a special case of an extended ring for a given ring R. Let {e_{1}, e_{2}, …, e_{n}} be a complete set of orthogonal primitive idempotents for R. Then for primitive idempotent e_{i} in R, we define R_{e_{i}} as follows;

(\begin{matrix}e_{1}Re_{1} & ⋯ & e_{1}Re_{i} & Y_{1} & e_{1}Re_{i+1} & ⋯ & e_{1}Re_{n} \\ ⋮ & ... & ⋮ & ⋮ & ⋮ \\ e_{i}Re_{1} & ⋯ & e_{i}Re_{i} & Y_{i} & e_{i}Re_{i+1} & ⋯ & e_{i}Re_{n} \\ X_{1} & ⋯ & X_{i} & U & X_{i+1} & ⋯ & X_{n} \\ e_{i+1}Re_{1} & ⋯ & e_{i+1}Re_{i} & Y_{i+1} & e_{i+1}Re_{i+1} & ⋯ & e_{i+1}Re_{n} \\ ⋮ & & ⋮ & ⋮ & ⋮ & ... \\ e_{n}Re_{1} & ⋯ & e_{n}Re_{i} & Y_{n} & e_{n}Re_{i+1} & ⋯ & e_{n}Re_{n} \end{matrix}) ,

where the X_{j} are e_{i}Re_{j} for j = 1, …, i − 1, i + 1, …, n, X_{i} is J(e_{i}Re_{i}), the Y_{k} are e_{k}Re_{i} for k = 1, …, n and U is e_{i}Re_{i}. Then R_{e_{i}} is a ring by usual matrix operations.

Remark 4. Proposition 8 says that a basic left Harada ring R has a Nakayama isomorphism if and only if so does R_{e} for e ∈ pi(R) = ⋃_{i=1}^{m}{e_{ij}}_{j=1}^{n(i)}.

We denote a basic ring of Q(R) by Q^{b}(R).

Remark 5. If R is a one-sided artinian QF-3 ring, the number of right S-primitive idempotents for R coincides with that of left S-primitive idempotents for R.

Theorem 9. Let R be a basic and left Harada ring and let Q = Q(R). Then Q has a Nakayama isomorphism if and only if so does R.

Proof. If Q is basic, then R = Q by Theorem 6. Hence we may assume that Q is not basic. Let pi(R) = ⋃_{i=1}^{m}{e_{ij}}_{j=1}^{n(i)} be a complete set of primitive idempotents for R as given in the proof of Theorem 6. Then if {e_{ij}}_{j=1}^{n(i)} has no right S-primitive idempotents, then e_{i1}Q ≅ e_{ij}Q for j = 2, …, n(i) by Lemma 5. If {e_{ij}}_{j=1}^{n(i)} has only one right S-primitive idempotent, say e_{ik}, then

e_{i1}Q ≅ e_{ij}Q  for j = 2, …, k;
e_{i,k+1}Q ≅ J(e_{ik}Q) and
e_{i,k+1}Q ≅ e_{ij}Q  for j = k + 2, …, n(i).

Moreover, if {e_{ij}}_{j=1}^{n(i)} has two right S-primitive idempotents, say, e_{ik}, e_{it} (k < t), then

e_{i1}Q ≅ e_{ij}Q  for j = 2, …, k;
e_{i,k+1}Q ≅ J(e_{ik}Q) and
e_{i,k+1}Q ≅ e_{ij}Q  for j = k + 2, …, t;
e_{i,t+1}Q ≅ J(e_{it}Q) and
e_{i,t+1}Q ≅ e_{ij}Q  for j = t + 2, …, n(i).

Repeating the same argument and Remark 5, we have the following sequences for i = 1, …, m;

e_{i1}Q > e_{i1}J(Q)
≀ ↑
e_{i,k_{1}+1}Q > J(e_{i,k_{1}+1}Q)
≀ ↑
e_{i,k_{2}+1}Q ⋯,

where e_{ik_{t}} is right S-primitive. Hence the complete set of the primitive idempotents pi(Q^{b}) for Q^{b} is ⋃_{i=1}^{m}{e_{i1}, e_{i,k_{t}+1}}_{t≥1} ⊆ pi(R) = pi(Q) and e_{i1} Q^{b} is injective. Since e_{i1} is left Sprimitive, e_{i1}R = e_{i1}Q by Remark 2 and so e_{i1}Re_{i1} = e_{i1}Qe_{i1}. Hence we have a ring isomorphism from Q^{b} to a subring of R.

(i) We choose {e_{h1}}_{h=1}^{n(h)} ⊂ pi(R) with e_{hn(h)} right S-primitive. We put e_{h} = e_{h1} + ⋯ + e_{hn(h)}. Then by Lemma 3 and Lemma 5, we e_{h}R = e_{h}Q. (ii) We choose {e_{h1}}_{h=1}^{n(h)} ⊂ pi(R) without right S-primitive. By Remark 3, Q_{e_{h1}}^{b} is isomorphism to a ring with the complete set ⋃_{i≠h}{e_{i1}, e_{i,k_{t}+1}}_{t≥1} ∪ {e_{h1}, e_{h2}} of primitive idempotents. Similarly repeating n(h) − 2 times, we can make an extended ring with the complete set ∪_{i≠h}{e_{i1}, e_{i,k_{t}+1}}_{t≥1} ∪{e_{hj}}_{j=1}^{n(h)} of primitive idempotents.

Letting,

Q^{b} = ( ∗  e_{11}Re_{h1}  ∗ ; ⋮ ; e_{h1}Re_{11} … e_{h1}Re_{h1} … e_{h1}Re_{m1} …; ⋮ ; ∗  e_{m1}Re_{h1}  ∗ ; ⋮) ,

Q_{e_{h1}}^{b} = (\begin{matrix}∗ & & e_{11}Re_{h1} & e_{11}Re_{h1} & & ∗ \\ & & ⋮ & ⋮ \\ e_{h1}Re_{11} & … & e_{h1}Re_{h1} & e_{h1}Re_{h1} & … & e_{h1}Re_{m1} & … \\ e_{h1}Re_{11} & … & J(e_{h1}Re_{h1}) & e_{h1}Re_{h1} & … & e_{h1}Re_{m1} & … \\ & & ⋮ & ⋮ \\ ∗ & & e_{m1}Re_{h1} & e_{m1}Re_{h1} & & ∗ \\ & & ⋮ & ⋮ \end{matrix}) .

For two submodules

A =  ( \begin{matrix}& 0 & & ⋯ & ⋯ & & & 0 \\ h1 > & e_{h1}Re_{11} & … & e_{h1}Re_{h1} & e_{h1}Re_{h1} & … & e_{h1}Re_{m1} & … \\ & 0 & & ⋯ & ⋯ & & & 0 \end{matrix} ),
B =  ( \begin{matrix}& 0 & & ⋯ & ⋯ & & & 0 \\ h1 > & 0 & & ⋯ & ⋯ & & & 0 \\ & e_{h1}Re_{11} & … & J(e_{h1}Re_{h1}) & e_{h1}Re_{h1} & … & e_{h1}Re_{m1} & … \\ & 0 & & ⋯ & ⋯ & & & 0 \end{matrix} )

of Q_{e_{h1}}^{b}, J(A) ≅ B by [13, Theorem 1].

Hence as a ring isomorphism,

(\begin{matrix}∗ & & e_{11}Re_{h1} & e_{11}Re_{h2} & & ∗ \\ & & ⋮ & ⋮ \\ e_{h1}Re_{11} & … & e_{h1}Re_{h1} & e_{h1}Re_{h2} & … & e_{h1}Re_{m1} & … \\ e_{h2}Re_{11} & … & e_{h2}Re_{h1} & e_{h2}Re_{h2} & … & e_{h2}Re_{m1} & … \\ & & ⋮ & ⋮ \\ ∗ & & e_{m1}Re_{h1} & e_{m1}Re_{h2} & & ∗ \\ & & ⋮ & ⋮ \end{matrix})

≅

(\begin{matrix}∗ & & e_{11}Re_{h1} & e_{11}Re_{h1} & & ∗ \\ & & ⋮ & ⋮ \\ e_{h1}Re_{11} & … & e_{h1}Re_{h1} & e_{h1}Re_{h1} & … & e_{h1}Re_{m1} & … \\ e_{h1}Re_{11} & … & J(e_{h1}Re_{h1}) & e_{h1}Re_{h1} & … & e_{h1}Re_{m1} & … \\ & & ⋮ & ⋮ \\ ∗ & & e_{m1}Re_{h1} & e_{m1}Re_{h1} & & ∗ \\ & & ⋮ & ⋮ \end{matrix})

by [13, Theorem 1] again.

(iii) We choose {e_{h1}}_{h=1}^{n(h)} ⊂ pi(R) with some right S-primitive idempotents. Then we denote a right S-primitive idempotent of {e_{h1}}_{h=1}^{n(h)} by e_{hk_{t}}. We reset

{e_{h1}}_{h=1}^{n(h)} = {e_{h1}, …, e_{hk_{1}}, …, e_{hk_{2}}, … }.

Then the complete set pi(Q^{b}) of Q^{b} is ⋃_{i=1}^{m}{e_{i1}, e_{i,k_{t}+1}}_{t≥1}. First by the same argument above for e_{i1}, e_{i,k_{1}+1}, we have a ring isomorphic to a ring with the complete set {e_{i1}, …, e_{i,k_{1}+1}} ⊂ pi(R). Next, by [13, Theorem 1], repeating the same argument like (ii), for e_{i,k_{1}+1}, e_{i,k_{2}+1}, we have a ring isomorphism to a ring with the complete set {e_{i1}, …, e_{ik_{1}}, e_{ik_{1}+1}, …, e_{ik_{2}}, e_{i,k_{2}+1}}. Hence the suitable extended ring of Q^{b} is isomorphic to R. Therefore, by Proposition 8, Q^{b} has a Nakayama isomorphism if and only if so does R. □

3. Another question

Oshiro’s result(Result A) in the introduction also poses another question whether there exist surjective ring homomorphisms:

Q(T_{1}) ↠^{ϕ̄_{1}} Q(T_{2}) ↠^{ϕ̄_{2}} … ↠^{ϕ̄_{n−1}} Q(T_{n}) ↠^{ϕ̄_{n}} Q(R)
∨  ∨  ∨  ∨
T_{1}  ↠^{ϕ_{1}} T_{2}  ↠^{ϕ_{2}} … ↠^{ϕ_{n−1}} T_{n}  ↠^{ϕ_{n}} R.

However K. Koike informed the author the following examples;

Example 3. Let Q be a local serial ring, and J(Q) ≠ 0, J(Q)^{2} = 0. Then J(Q) = S(Q). We put

R = (\begin{matrix}Q & Q \\ J & Q \end{matrix}) / (\begin{matrix}0 & J \\ 0 & J \end{matrix}) ,

where J = J(Q). Then R is a serial ring of an admissible sequence (3,2) and so we see that R = Q(R). Also

T_{1} = (\begin{matrix}Q & Q \\ J & Q \end{matrix}) , T_{2} = (\begin{matrix}Q & Q \\ J & Q \end{matrix}) / (\begin{matrix}0 & J \\ 0 & 0 \end{matrix}) ,

Q(T_{1}) = (\begin{matrix}Q & Q \\ Q & Q \end{matrix}) , Q(T_{2}) = T_{2}.

(\begin{matrix}J & J \\ J & J \end{matrix}) is a unique non-trivial ideal of Q(T_{1}). Hence there does not exist a surjective

ring homomorphism Q(T_{1}) to Q(T_{2}).

Example 4. We put

T = (\begin{matrix}K & K & K \\ 0 & K & K \\ 0 & 0 & K \end{matrix}) , I = (\begin{matrix}0 & 0 & K \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) ,

where K is a field, and R = T/I. Then R is a serial ring of an admissible sequence (2,2,1) and we have a natural map

T = T_{1} → R.

However the maximal quotient ring Q(T) of T is the full matrix algebra with degree 3 over a field K and Q(R) = R. Since Q(T) is semisimple, there does not exist a surjective ring homomorphism Q(T) to Q(R).

References

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