ON HARADA RINGS AND

SERIAL ARTINIAN RINGS

Thanakarn Soonthornkrachang†, Phan Dan∗,

Nguyen Van Sanh∗∗, and Kar Ping Shum‡

∗∗† Department of Mathematics Mahidol University, Bangkok 10400, Thailand e-mail: frnvs@mahidol.ac.th

∗ Department of Mathematics University of Transport of Ho Chi Minh City, Vietnam e-mail: mrphandan@yahoo.com

‡Faculty of Science The Chinese University of Hong Kong, Hong Kong SAR e-mail: kpshum@math.cuhk.edu.hk

Abstract

A ring R is called a right Harada ring if it is right Artinian and every non-small right R-module contains a non-zero injective submodule. The first result in our paper is the following: Let R be a right perfect ring. Then R is a right Harada ring if and only if every cyclic module is a direct sum of an injective module and a small module; if and only if every local module is either injective or small. We also prove that a ring R is QF if and only if every cyclic module is a direct sum of a projective injective module and a small module; if and only if every local module is either projective injective or small. Finally, a right QF-3 right perfect ring R is serial Artinian if and only if every right ideal is a direct sum of a projective module and a singular uniserial module.

Key words: Harada ring, artinian ring, small module, co-small module, AMS (2000) Mathematics Subject Classifications: 16D50, 16D70, 16D80 ‡ Partially supported by USC grant (HK)# 2160187 ∗∗ Corresponding author

1. Introduction and Preliminaries

Throughout this paper, all ring are associate rings with identity and all right R-modules are unitary. For a right R-module M, we denote E(M), J(M) and Z(M) the injective hull, radical and the singular submodule of M. Especially, J(R) is the Jacobson radical of the ring R. A right R-module M is called uniserial if the lattice of its submodules is linear. Call M a serial module if it is a direct sum of uniserial modules. A ring R is right serial if R_{R} is serial as a right R-module. A ring R is serial if it is right and left serial. Call a ring R serial Artinian if it is serial and two-sided Artinian.

Call M a small module if M is small in E(M) otherwise, we call a nonsmall module. Dually, a right R-module M is called a co-small module if for any epimorphism from P to M with P is projective, ker(f) is essential in P. A non-cosmall module is defined as a not co-small module. A right R-module is a local module if it has the greatest proper submodule. A ring R is called a Quasi-Frobenius ring (briefly QF-ring) if it is right self-injective right Artinian. Call a ring R a right QF-3 ring if E(R_{R}) is projective (see [14] and [21]).

Manabu Harada [9], [10] and [11] has studied some generalizations of QFrings by introducing the following two conditions:

(*) Every non-small right R-module contains a non-zero injective submodule;

(**) Every non-cosmall right R-module contains a non-zero projective direct summand.

It should be noted that right perfect rings with (*) and semiperfect rings with (**) are characterized in terms of ideals in [11],[12] and [13]. Oshiro [16] gave some characterizations of these kinds of rings and introduced the definitions of right Harada and right co-Harada rings as follows.

A ring R is called a right Harada ring if it is right Artinian and (*) holds. A ring R is a right co-Harada ring if it satisfies (**) and ACC on right annihilators.

In this paper, we give some characterizations of right Harada rings and serial Artinian rings. In section 1, we recall some well-known results which will be used in this paper. We characterize the classes of right Harada rings and QF-rings by right perfect rings and cyclic or local modules. Section 3 is concerned with serial Artinian rings. For convenience, we list some well-known results here to use in this paper.

Theorem A ([16, Theorem 2.11]) For a ring R, the following conditions are equivalent:

(1) R is a right Harada ring;

(2) R is right perfect and satisfies the condition that the family of all projective modules is closed under taking small covers, i.e., for any exact sequence P →^{φ} E → 0, with E is injective and P is projective, ker(φ) is small in P;

(3) Every right R-module is a direct sum of an injective module and a small module;

(4) Every injective module is a lifting module.

Theorem B ([11, Theorem 3.6]) Let R be a semiperfect ring and {e_{i}}_{i=1}^{n} ∪ {f_{j}}_{j=1}^{m} a complete set of orthogonal primitive idempotents of R, where each e_{i}R is non-small, i = 1, … n, and f_{j}R is small, j = 1, … m. Then (**) holds if and only if:

(a) n ≥ 1 and e_{i}R is injective, i = 1, …, n;

(b) For each f_{j}, there exists e_{i} such that f_{j}R can be embedded in e_{i}R;

(c) For each f_{j}, there exists an integer n_{j} such that e_{i}J^{t} is projective for 0 ≤ t ≤ n_{i} and e_{i}J^{n_{i}+1} is a singular module, where J = J(R);

Further in this case, it is shown that every submodule e_{i}B in e_{i}R is either contained in e_{i}J^{n_{i}+1} or equal to some e_{i}J^{i}.

Theorem C ([16]) Let R be a right Artinian ring. The R is a serial ring if and only if for any primitive idempotent f of R, the injective hull E(fR) is a uniserial module.

2 Right Harada rings

In this section, we will prove that the classes of right Harada rings and QFrings are both characterized by perfect rings and cyclic (or local) modules. The proof of the following Lemma is routine and therefore is omitted.

Lemma 1 If {X_{i}, i = 1, …, n} is a family of small modules, then X = ∑_{i} X_{i}

is also small.

Proposition 2 Let R be a right perfect ring. Then the following conditions are equivalent:

(1) The condition (*) holds;

(2) Every non-small cyclic module contains a non-zero injective module;

(3) Every local module is either injective or small.

Proof (1) ⇒ (2) is clear. We now prove (2) ⇒ (3). Let M be a local module. Since R is right perfect, it follows from [5, Proposition 18.23] that there exists a primitive idempotent g of R such that M ≅ gR/G, G ⊂ gR. Hence M is an indecomposable cyclic module. Therefore M either injective or small by (2).

(3) ⇒ (1) Let M be a non-small module and E = E(M). Since R is right perfect, it follows that M ⊄ EJ (since EJ is small in E). Take any m ∈ M \ EJ. Then mR is a non-small module. Let {e_{i}|i = 1, 2, …, n} be an

orthogonal system of primitive idempotents of R. Then mR = ∑_{i=1}^{n} me_{i}R. By

Lemma 1, there exists an idempotent e_{i} such that me_{i}R is non-small. Since me_{i}R ≠ 0 and me_{i}R ≅ e_{i}R/H, H ⊂ e_{i}R, the module me_{i}R is local. It follows from (3) that me_{i}R is injective. Thus M contains a non-zero me_{i}R, hence (∗) holds. □

Theorem 3 Let R be a right perfect ring. Then the following conditions are equivalent:

(1) R is a right Harada ring;

(2) Every cyclic module is a direct sum of an injective module and a small module;

(3) Every local module is either injective or small.

Proof (1) ⇒ (3) By Theorem A.

(2) ⇒ (3) By the same argument as that of (2) ⇒ (3) in the proof of Proposition 2.

(3) ⇒ (1) By Proposition 2, R satisfies (∗). Hence by [12, Theorem 5], R is a right Artinian ring, and therefore it is a right Harada ring. □

Theorem 4 Let R be a right perfect ring. Then the following conditions are equivalent:

(1) R is a QF-ring;

(2) Every cyclic module is a direct sum of a projective injective module and a small module; (3) Every local module is either projective injective or small.

Proof (1) ⇒ (2) Since R is QF, it is a right Harada ring by [16, Theorem 4.3]. For a right R-module M, we have M = I ⊕ S, with I is injective ans S is small by Theorem A. It is clear that I is projective, proving (2).

(2) ⇒ (3) It follows from Proposition 2.

(3) ⇒ (1) It suffices to show that every injective module is projective. Since R is a right perfect ring satisfying (3), in view of Theorem 3, R is a right Harada ring. Let Q be an injective module. Then Q has a decomposition Q = ⨁_{I} Q_{i}, where each Q_{i} is a non-zero indecomposable module. We will show

that each Q_{i} is projective for each i ∈ I. Since R is right Artinian, Q_{i} contains a maximal submodule Q′_{i} (see [1, Theorem 28.4]). Let X be a proper submodule of Q_{i} such that X ⊄ Q′_{i}. Then Q′_{i} + X = Q_{i} = E(Q′_{i}). Hence Q′_{i} is a non-small module. Since R is a right Harada ring, it implies that Q′_{i} contains a proper direct summand, a contradiction. Thus Q′_{i} is the greatest proper submodule of Q_{i}, i.e., Q_{i} is a local module. Hence, by (3), Q_{i} is projective. It follows that Q is a projective module, proving that R is QF.

3. Characterizations of serial Artinian rings

First we recall a remark due to M. Harada in [11] as folllows:

Remark 5 Let R be a right perfect ring. Then R has a decomposition of the form:

R = ⨁_{i=1}^{n} e_{i}R ⊕ ⨁_{m=1}^{m} f_{j}R

where {e_{i}, i = 1, …, n} ∪ {f_{j}, j = 1, …, m} is the set of mutually orthogonal idempotents with each e_{i}R is non-small and f_{j}R is a small module, and always we have n ≥ 1.

Lemma 6 Let R be a right perfect ring and M a uniserial right R-module. Then every submodule of M is cyclic and hence M is a Noetherian module.

Proof Let N be a non-zero submodule of M. Since R is right perfect, it follows from [1, Theorem 28.4] that N contains a maximal submodule N′. Take any x ∈ N \ N′. Then xR ⊄ N′. Since M is uniserial, we must have N′ ⊂ xR. Hence N = xR, proving that M is a Noetherian module. □

Lemma 7 Let R be a right perfect ring. If E(gR) is uniserial for any primitive idempotent g of R, then R is serial Artinian.

Proof By Theorem C, it is enough to show that R is right Artinian. Since R is right perfect, we can write

R = ⨁_{i=1}^{n} g_{i}R

where {g_{i}, i = 1, … n} is a system of orthogonal primitive idempotents. For each i, E(g_{i}R) is uniserial, therefore by Lemma 6, it is Noetherian and hence R is right Noetherian. Combining with the assumption that R is right perfect, it follows that R is right Artinian by [1, Corollary 15.23] and this completes our proof. □

Theorem 8 Let R be a right perfect ring. Then the following conditions are equivalent:

(1) R is a serial Artinian ring;

(2) Every cyclic module is a direct sum of an injective module and a uniform small module;

(3) Every local module is either injective or uniform small.

Proof (1) ⇒ (2) Let R be a right perfect serial ring. Then by [16, Theorem 4.5], R is a right Harada ring. For a cyclic module M, we have M = I ⊕ S, where I is injective and S is small (see Theorem A). Since R is serial Artinian, S = ⨁_{j} S_{j} with S_{j} is uniserial. It follows from [11, Lemma 1.1] that S_{j} is small,

proving (2).

(2) ⇒ (3) by Proposition 2.

(3) ⇒ (1) Supppose that R has a decomposition as in Remark 5. Then by (3), e_{i}R is injective for each e_{i} ∈ {e_{i}, i = 1, …, n}. Hence by [11, Theorem 1.3], the ring R is right QF-3. It follows that E(f_{k}R) is projective for each

f_{k} ∈ {f_{j}, j = 1, …, m}. Therefore E(f_{k}R) = ⨁_{i∈I,j∈J} e_{ij}R, e_{ij}R ≅ e_{i}R, I ⊂

{1, ….n}. Since f_{k}R is uniform by (3), it implies E(f_{k}R) ≅ e_{i}R for some idempotent e_{i}. Next, we will show that e_{i}R is uniserial for all e_{i}, i = 1…. , n.

Let U, V be non-zero submodules of eR, where e ∈ {e_{i}, 1 ≤ i ≤ n}. Put I = U ∩ V. Assume that I ≠ U and I ≠ V. Then the module B = eR/I is not uniform, and hence B is not injective because it is indecomposable. Thus the local module B is neither uniform, nor injective, this contradicts to the assumption (3). Therefore, either I = U or I = V, proving that eR is a uniserial module. In view of Lemma 7, it follows that R is a serial Artinian ring. □

Lemma 9 ([11],[19]) The following statements hold for non-cosmall modules:

(1) A right R-module M is non-cosmall if it does not coincide with its singular submodule;

(2) If an R-module M contains a non-zero projective submodule, then it is non-cosmall.

Lemma 10 Let R be a right QF-3 semiperfect ring. If every uniform principal right ideal of R is either projective or singular as a right R-module, then R satisfies two conditions (a) and (b) of Theorem B.

Proof Since R is semiperfect, it has a decomposition of the form

R = e_{1}R ⊕ e_{2}R ⊕ ⋯ ⊕ e_{n}R,

where {e_{i} 1 ≤ i ≤ n} is the set of mutually orthogonal primitive idempotents. Since R is right QF-3 (i. e., E(R_{R}) is projective), it follows that there exists at least one e_{i} such that e_{i}R is injective. Without lost of generality we may assume that e_{i}R is injective for 1 ≤ i ≤ k and e_{j}R is not injective with k + 1 ≤ j ≤ n. take e = e_{j}, k + 1 ≤ j ≤ n. Since E(R_{R}) is projective, so is E(eR). Hence

E(eR) = ⨁_{i=1}^{k} ⨁_{t=1}^{t(i)} e_{it}R

where e_{it}R ≅ e_{i}R for 1 ≤ t ≤ t(i). Put

Q = ⨁_{i=1}^{k} ⨁_{t=1}^{t(i)} e_{it}R

and let α : E(eR) → Q be an isomorphism and π_{it} : ⨁_{i=1}^{k} ⨁_{t=1}^{t(i)} e_{it}R ⟶ e_{it}R the

projections. Let F = α(eR) and F_{it} = π_{it}(F) ⊂ e_{it}R ≅ e_{i}R. Then F_{it} is cyclic. It is easy to see that F_{it} is isomorphic to a principal right ideal of R. Moreover, F_{it} is uniform. Hence by hypothesis, F_{it} is either projective or singular for all pairs (i, t).

Suppose that F_{it} are singular for all pairs (i, t). It follows that F is singular, since F ⊂ ⨁_{i} F_{it}, which is a contradiction to the fact that F ≅ eR. Therefore

there exists a pair (i_{0}, t_{0}) such that F_{i_{0}t_{o}} is non-zero projective. Let p = π_{i_{0}t_{0}}|_{F} , induced by the projection, and consider the exact sequence F →^{p} F_{i_{0}t_{o}} → 0 with F_{i_{0}t_{0}} is projective. Then F = kerp ⊕ F′, for some F′ ≅ F_{i_{0}t_{0}}. Since F (≅ eR) is indecomposable and F′ ≠ 0, it implies that kerp = 0, hence F ∩ ⨁_{i≠i_{0},t≠t_{0}} e_{it}R = 0, because F ⊂ eQ ( i.e., F is essential in Q). Therefore

Q = e_{i_{0}t_{0}}R. Thus eR ≅ F ⊂ e_{i_{0}t_{0}}R = e_{i_{0}}R. Hence R satisfies both conditions (a) and (b) of Theorem B and our Lemma has been proved. □

Lemma 11 ([11, Theorem 3.6]) Let R be a semiperfect ring and e a primitive idempotent of R such that eR is injective. Suppose that every submodule of eR is either projective or singular. Then there exists an integer n such that eJ^{t} is projective for 0 ≤ t ≤ n and eJ^{n+1} is singular, where J is the Jacobson radical of R.

Lemma 12 Let R be a right QF-3 right perfect ring and e ∈ R a primitive idempotent of R such that eR is injective. If every 2-generated right submodule of eR is either projective or singular, then every submodule of eR is either projective or singular.

Proof It is clear that eR is uniform. Let Z(eR) be the singular submodule of eR. We first prove that eR/Z(eR) is uniserial. Suppose on the contrary that there are submodules U and V of eR such that Z(eR) ⊂ U ∩ V and U ⊄ V, V ⊄ U. Take u ∈ U \ V and v ∈ V \ U. Consider the module X = uR + vR. Since X is a 2-generated right submodule of eR, by assumption, it is projective and singular. But both uR and vR are not singular, X must be not singular. Hence X is projective. Let X_{1} and X_{2} be maximal submodules of uR and vR respectively. Then X_{1}+vR and X_{2}+uR are distinct maximal submodules of X. This contradicts to the fact that X is a projective indecomposable module on a right perfect ring. It would imply that eR/Z(eR) is a uniserial module. We now show that eR/Z(eR) has finite length. Let X_{1} be the largest submodule of eR. Since R is right perfect, it follows from [1, Theorem 28.4] that X_{1} (here, X_{1} = rad(eR)) contains a maximal submodule X_{2}. Continuing this process we get a strictly descending chain

eR ⊃ X_{1} ⊃ X_{2} ⊃ ⋯ ⊃ X_{n} ⊃ ⋯ ⊃ Z(eR).

We will prove that this chain is stationary. It is clear that each X_{i} is a local module, hence each X_{i} is an epimorphism image of fR for some primitive idemppotent f of R. Since R is right QF-3, it follows that fR is uniform, and hence X_{i} is projective or singular. But X_{i} ⊃ Z(eR), and X_{i} ≠ Z(eR), it follows that each X_{i} is projective. We claim that for i ≠ j, X_{i} ≅ X̸_{j}.

Suppose on the contrary that there is an isomorphism φ : X_{i} → X_{j}. Since eR is injective, φ can be extended to φ̄ : eR → eR and φ̄ is also an isomorphism. From this we obtain eR/X_{i} ≅ eR/X_{j}, and this contradicts to the fact that length(eR/X_{i}) ≠length (eR/XC_{j}). Since the representative set of R is finite, it implies that the chain X_{1} ⊃ X_{2} ⊃ … must be stationary. Therefore the condition (c) of theorem B is satisfied, proving our Lemma. □

Theorem 13 Let R be a semiperfect ring. Then the following conditions are equivalent:

(1) (**) holds;

(2) R is right QF-3 and every right ideal is a direct sum of a projective module and a singular module.

(3) R is right QF-3 and every uniform right ideal is either projective or singular.

Proof (1) ⇒ (2). In view of Theorem B, we see that E(R_{R}) is projective, i.e., R is right QF-3. Moreover, R has finite right Goldie dimension. Let B be a right ideal of R. If B is non-cosmall, then by (**), we have B = B_{1} ⊕ B′_{1} with B_{1} is non-zero and projective.

Again, if B′_{1} is non-cosmall, then B′_{1} = B_{2} ⊕ B′_{2}, with B_{2} is non-zero and projective. Since R_{R} has finite Goldie dimension, we get B_{i} is finite dimensional and therefore after a finite number of steps, we get B = B_{1} ⊕ ⋯ ⊕ B_{k} ⊕ B′_{k}, where B_{1}, …, B_{k} are projective and B′_{k} is cosmall, i.e., singular (Lemma 9). Hence (2) holds.

(2) ⇒ (3). Obvious.

(3) ⇒ (1). By Lemmas 10 and 11. □

Theorem 14 Let R be a right perfect ring. The following conditions are equivalent:

(1) R satisfies (**);

(2) R is right QF-3 and every 2-generated right ideal is a direct sum of a projective module and a singular module; (3) R is right QF-3 and every uniform 2-generated right ideal is either pro-

jective or singular.

Proof The proof of (1) ⇒ (2) is similar to that of Theorem 13. The implication (2) ⇒ (3) is obvious. From Lemmas 10 and 12, it follows that R satisfies three conditions (a), (b) and (c) of Theorem B. Therefore, R satisfies (**), proving (3) ⇒ (1). □

Theorem 15 The following conditions are equivalent for a right QF-3 semiperfect ring R.

(1) R is a serial Artinian ring;

(2) Every right ideal B of R has a decomposition of the form B = B_{0} ⊕ ⨁_{1≤i≤n} B_{i}, with B_{0} is projective and each B_{i}(1 ≤ i ≤ n) is a singular

uniserial module of finite length;

(3) Every right ideal of R is either a projective module or a singular uniserial module of finite length.

Proof (1) ⇒ (2). Let R be a serial Artinian ring. Then R is a right co-Harada ring by [16, Theorem 4.5]. Hence for any right ideal B of R, B = B_{0}⊕B′, where B_{0} is projective and B′ is singular by [16, Theorem 3.18]. Then B′ = ⨁_{i∈I} B_{i},

where each B_{i} is uniserial with finite length, proving (2).

(2) ⇒ (3). Obviously.

(3) ⇒ (1). Since R is right QF-3 and semiperfect, it follows from Theorem 13 that R satisfies the condition (**). By applying Theorem B, R has a decomposition

R = ⨁_{i=1}^{n} e_{i}R ⊕ ⨁_{j=1}^{n} f_{j}R,

where e_{i}R is injective and f_{j}R is small. Moreover for each j we have E(f_{j}R) ≅ e_{i}R for some i. By Theorem C, in order to prove that R serial, it suffices to show that R is right Artinian and each e_{i}R is uniserial, i ∈ {1, 2, …, n}. Take any e ∈ {e_{i}, i = 1, … n} and consider the module eR with its singular submodule Z = Z(e). By Theorem B (3), Z is a uniserial module with finite length. Therefore, eR is an Artinian module. On the other hand, again by Theorem B, we have either eB ⊂ Z or Z ⊂ eB. Since Z and eR/Z are both uniserial, it follows that eR is a uniserial module. Moreover, eR/Z is of finite length. It is now clear that R is right Artinian and E(eR) is uniserial for any primitive idempotent e of R. The proof is now complete. □

Theorem 16 Let R be a right QF-3 right perfect ring. The following conditions are equivalent:

(1) R is serial Artinian;

(2) Every right ideal of R is a direct sum of a projective module and a singular uniserial module.

(3) Every uniform 2-generated right ideal is either projective or singular uniserial.

Proof The proof of (1) ⇒ (3) is similar to that of Theorem 15 and (2) ⇒ (3) is obvious. We now prove that (3) ⇒ (1). Clearly, R satisfies (**) by Theorem 14. By Remark 5, we can write R in the form:

R = ⨁_{i=1}^{n} e_{i}R ⊕ ⨁_{j=1}^{m} f_{j}R

with properties in the Remark 5.

By Theorem B, we can see that each e_{i}R is injective and for each j, we have E(f_{j}R) ≅ e_{i}R for some i = 1, …, n. Using Lemma 7, we now show that each e_{i}R is a uniserial module, i = 1, …, n. Put Z = Z(e_{i}R). Since R satisfies (**), by the same way as in Theorem 15, we can see that e_{i}R/Z is a uniserial module with finite length, and for every right ideal B of R, we have either Z ⊂ e_{i}B or e_{i}B ⊂ Z. Therfore, it suffices to show that Z is also a uniserial module. We can suppose that Z ≠ 0.

Let U, V be non-zero submodules of Z. If U ⊄ V and V ⊄ U, we can take any u ∈ U \ V and v ∈ V \ U, and consider the module C = uR + vR. Then C is a uniform 2-generated right ideal of R and C is singular. However C is not uniserial, since uR ⊄ vR and vR ⊄ uR, and this contradicts to the hypothesis (3). Hence, either U ⊂ V or V ⊂ U, proving that Z(eR) is uniserial. The proof is now complete. □

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