A Note on Co-Harada Rings

Abstract

In the early 1990s, Harada and Oshiro introduced extending and lifting properties for modules and, simultaneously, considered two new classes of artinian rings which contain quasi-Frobenius (QF-) rings and Nakayama rings: one is the class of right Harada rings and the other is the class of right co-Harada rings. Although QF-rings and Nakayama rings are left-right symmetric, Harada and co-Harada rings are not left-right symmetric. However, Oshiro showed that left Harada rings and right co-Harada rings are coinside. In this paper we provide many characterizations of right co-Harada rings and (right and left) co-Harada rings.

1 Introduction

Throughout this paper all rings will be associative with identity and modules will be unital modules. For an R-module M we write M_R (_RM) to indicate that M is a right (left) R-module. By J(M), E(M), Z(M) we denote the Jacobson radical, the injective hull and the singular submodule of M, respectively. We denote the set of primitive idempotents of R by Pi(R). A ring R is said to have enough idempotents if the identity element of R can be written as the sum of a finite number of orthogonal primitive idempotens of R.

Let R be a ring and M a right R-module. N ≤ M will mean N is a submodule of M. A submodule N of M is called small in M, denoted by N ≤_smM, whenever for every submodule L of M, N + L = M implies L = M. A non-zero submodule N of M is said to be an essential submodule of M, denoted by N ≤_eM, if for every 0≠L ≤ M, N ∩ L≠ 0. A non-zero module M is called uniform if N ≤_eM for every non-zero submodule N of M.

A module M is said to be small if M is small in its injective hull. A (right) R-module M is called non-small if M is not a small submodule in its injective hull, which is equivalent to the fact that M is not a small submodule in any extension module of M (see [11, Proposition 1.1]). Dually, M is called a non-cosmall module, following [19], if M is a homomorphic image of a projective module P whose kernel is not essential in P, which is equivalent to the fact that if M is a homomorphic image of a module N, then the kernel is always not essential in N (see [11, Proposition 3.1]).

A module M is called an extending module (CS module) if every submodule of M is essential in a direct summand of M. A ring R is called right CS if R_R is an extending module. A module M is said to be a local module if M has a unique maximal submodule.

A ring R is called right QF-3 if R_R has a direct summand eR (e is an idempotent of R) which is a faithful injective right ideal, and it is called right QF-3⁺ if E(R_R) is projective. A ring R is called right QF-2 if it is a direct sum of right uniform ideals (as right R-modules). R is called right (left) nonsingular if the right (left) singular ideal of R is zero.

M is called uniserial if the set of submodules of M is linearly ordered and M is called serial if it is a direct sum of uniserial modules. A ring R is a right (left) serial ring if R_R (_RR) is a serial module, and R is called a serial ring if R are both right and left serial. A two-sided artinian serial ring is also called a Nakayama ring.

A submodule N of a module M is called a waist in M if either N ≤ X or X ≤ N is satisfied for any submodule X of M.

Definition 1

A module M is called z-serial if M satisfies three following conditions:

• (1) M is uniform,

• (2) Z(M) is a waist in M,

• (3) M/Z(M) is uniserial.

The following series of results from various sources is presented here in order to make it easier to refer to them later in the paper.

Lemma 1

([7, Lemma 7.1]) Every direct summand of an extending module is an extending module.

Lemma 2

([9, 18.23]) Every local right (left) R-module over semiperfect ring R is isomorphic to a homomorphic image of eR (Re) for some e ∈Pi(R). If R is right (left) QF-2 then every local right (left) R-module is either projective or singular.

In [11] Harara has studied the following conditions:

(∗)_r:

Every non-small right R-module contains a non-zero injective submodule.

$(*)^{*}_{r}$ :

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

He also gave a characterization of semiperfect rings with $(*)^{*}_{r}$ as follows:

Theorem 1

([11, Theorem 3.6]) Let R be semiperfect. Then $(*)^{*}_{r}$ holds if and only if there exists a set of primitive idempotents {e_i} and of integers {n_i} such that:

• (1) e_iR is injective,

• (2) $e_{i}J^{t_{i}}$ is projective for t_i ≤ n_i and $e_{i}J^{n_{i}+1}$ is singular, and

• (3) Every indecomposable projective module is isomorphic to some $e_{i}J^{t_{i}}$.

In this case, every submodule e_iB in e_iR either is contained in $e_{i}J^{n_{i}+1}$ or equal to some $e_{i}J^{t_{i}}$, t_i ≤ n_i + 1, where J = J(R).

The following classes of rings have been defined by Oshiro [16]: A ring R is called a right Harada ring if it is right artinian and satisfies the condition (∗)_r. Dually, a ring R is called a right co-Harada ring if it satisfies the condition $(*)^{*}_{r}$ and the ACC on right annihilators.

Huynh in [13] studied right co-Harada rings under the name right Σ-CS rings. Many results on onesided Harada (or co-Harada) rings are given in [1, 10] and [16].

In 1993, Vanaja ([21]) has generalized $(*)^{*}_{r}$ by considering the following condition

$(*)^{*}_{1,r}$ :

Every finitely generated non-cosmall right R-module contains a non-zero projective direct summand.

It is known ([21, Theorem 1.10]) that the following are equivalent for a semiperfect ring R: (1) R satisfies $(*)^{*}_{1,r}$; (2) $R_{R}^{(n)}$ is an extending module; and (3) Direct sum of any two indecomposable projective right R-modules is extending.

Lemma 3

([9, Theorem 20.15]) Every indecomposable injective and projective right R-module M is isomorphic to a summand of R, that is, there exists an idempotent e ∈ R such that M≅eR.

Lemma 4

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

• (1) An R-module M is non-cosmall if and only if M≠Z(M);

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

From the definition of non-cosmall modules and Lemma 3 we have

Lemma 5

The following statements are equivalent for a ring R and a cardinal α:

• (1) $R_{R}^{(\alpha )} = {\oplus }_{I}R_{R}$ is an extending module, where card(I) = α.

• (2) Every α-generated right R-module M is a direct sum of a projective module and a singular module.

The proof of the following lemma is straightforward and will be omitted.

Lemma 6

([11]) Let R be a ring.

• (1) If $\{X_{i}\}^{n}_{i=1}$ is a set of small submodules of a right R-module X, then ${\sum }_{i=1}^{n}X_{i}$ is a small module,

• (2) If R is right perfect, then a right R-module M is nonsmall if and only if there exists an element m ∈ M such that module cyclic mR is nonsmall.

Lemma 7

([17]) A ring R is a right (resp. left) Harada ring if and only if R is left (resp. right) co-Hadada.

The following lemma is a special case of [2, Lemma 11].

Lemma 8

Let U be a uniform module, and suppose that U is not isomorphic to any of its proper submodules. Then End(U) is a local ring.

Proof

Let α ∈End(U). Then Ker(α) ∩Ker(1 − α) = 0 in U. Since U is uniform, it follows that either Ker(α) = 0 or Ker(1 − α) = 0, respectively either α or (1 − α) is monomorphic. Since U is not isomorphic to any of its proper submodules, one of two homomorphisms is isomorphic, as required. □

2 Co-Harada Rings

We start the section with a lemma which will be useful in the sequel.

Lemma 9

Let M be a uniform cyclic module such that E(M)/M has ACC on cyclic submodules. Then M is not isomorphic to any of its proper submodules, and hence End(M) is a local ring.

Proof

Let E = E(M) and suppose that there is a proper submodule U of M and an isomorphism $\varphi : U \rightarrow M$. Consider the commutative diagram

where $\bar {\varphi }$ is an extension of φ. Let $\bar {\varphi }(eR)$ = U₁ and $\bar {\varphi }(U_{1})$ = U₂, and $\bar {\varphi }(U_{n})$ = U_n+ 1 for any integer number n. Since E is uniform and φ is an isomorphism, it follows that $\bar {\varphi }$ is a monomorphism. So that, we get an infinite strictly ascending chain of cyclic modules M ≤ U₁ ≤ U₂ ≤⋯ ≤ U_n ≤⋯ in E, which is a contradiction, since E/M has ACC for cyclic submodules. Hence, M is not isomorphic to any of its proper submodules. Moreover, by Lemma 8, End(M) is local. □

The following theorem provides characterizations of semiperfect rings with $(*)^{*}_{r}$. This generalizes [15, Theorem 2.3] and [20, Theorem 3.8] (without assuming E(eR) is a z-serial module for every e ∈Pi(R)).

Theorem 2

Let R be a ring. The following statements are equivalent:

• (1) R is a semiperfect ring with $(*)^{*}_{r}$.

• (2) R is a ring with enough idempotents, eR is a waist in E(eR) and E(eR)/eR has ACC on cyclic submodules for every e ∈Pi(R).

• (3) R is a ring with enough idempotents, eR is a waist in E(eR) and eR/Z(eR) has finite length for any e ∈Pi(R).

Proof

(1) ⇒ (2). It is obvious, see Theorem 1.

(2) ⇒ (1). Assume (2). Then R_R = e₁R ⊕ e₂R ⊕⋯ ⊕ e_nR, where e_i ∈Pi(R) and e_iR is indecomposable. Let e = e_i. Since eR is a waist in E(eR), it follows that eR is uniform, so that End(eR) is local by Lemma 9. Hence R is a semiperfect ring. By [20, Theorem 2.4], R satisfies condition $(*)^{*}_{1,r}$. Moreover, every eR is not isomorphic to any of its proper submodules, by Lemma 9. So that R satisfies $(*)^{*}_{r}$, by [20, Lemma 3.1].

(3) ⇒ (1) Assume (3). Then R_R = e₁R ⊕ e₂R ⊕⋯ ⊕ e_nR, where e_i ∈Pi(R) and e_iR is indecomposable. Let e = e_i. It follows that eR is uniform, since eR is a waist in E(eR). If Z(eR) = 0 then End(eR) is local, since eR has finite length. If eR is injective then End(eR) is local, since eR is uniform. We consider the case W = Z(eR)≠ 0 and E = E(eR)≠eR. By (3), length(eR/W) = n > 0. Using the same argument as in the proof of Lemma 9, we get an infinite strictly ascending chain of cyclic modules eR ≤ U₁ ≤ U₂ ≤⋯ ≤ U_n ≤⋯ in E, where eR≅U_i for $i = 1,2,\dots$. Since Z(E)≠E and eR is a waist in E, Z(U_i)≠eR, so that Z(U_i) = Z(eR) = W. Then eR/W is not a module with finite length, a contradiction. It follows that eR is not isomorphic to any of its proper submodules. Hence End(eR) is local, by Lemma 8. Therefore R is a semiperfect ring. Since eR is a waist in E(eR) for every e ∈Pi(R), R satisfies $(*)^{*}_{1,r}$, by [20, Theorem 2.4]. Since every eR is not isomorphic to any of its proper submodules, it follows R satisfies $(*)^{*}_{r}$, by [20, Lemma 3.1]. □

A well-known result of C. Faith [8] asserts that a right self-injective ring is QF if and only if R has ACC (or DCC) on right annihilators. Next we provide some similar characterizations of right co-Harada rings.

Theorem 3

Let R be a ring satisfying the ACC (or DCC) on right annihilators. Then the following statements are equivalent:

• (1) R is a right co-Harada ring;

• (2) $R^{(2)}_{R}$ is an extending module and E(R_R)/R has ACC on cyclic submodules.

• (3) For every e ∈Pi(R), eR is a waist in E(eR) and E(eR)/eR has ACC on cyclic submodules.

• (4) For every e ∈Pi(R), eR is a waist in E(eR), E(eR) is a projective module and eR/Z(eR) has ACC on cyclic submodules.

Proof

(1) ⇒ (2). It is obvious.

(2) ⇒ (3). Assume (2). It is easy to see that R is a direct sum of indecomposable right ideals if R satisfies the ACC (or DCC) on right annihilators. Since $R^{(2)}_{R}$ is extending, so is R_R. Hence R is a direct sum of uniform right ideals. Let R_R = e₁R ⊕ e₂R ⊕⋯ ⊕ e_nR, where $\{e_{i}\}_{i=1}^{n}$ is a set of primitive idempotents of R and each e_iR is uniform. Let e = e_i and E = E(e_iR). It is easy to see that E/eR has ACC for cyclic submodules, since it is isomorphic to a direct summand of E(R_R)/R. If eR is injective then End(eR) is local since eR is uniform. Assume eR is not injective. It follows from Lemma 9 that End(eR) is local. So that R is semiperfect. Since ${R_{R}^{2}}$ is extending, it follows from [20, Lemma 12.8] that eR is a waist in E(eR). Therefore (3) holds.

(3) ⇒ (1). Assume (3). If R has the ACC (or DCC) on right annihilators then R is a ring with enough idempotents. It follows from (3) and Theorem 2 that R is a semiperfect ring with $(*)^{*}_{r}$. Hence R is a right co-Harada ring, by [6, Corollary 3.8].

(1) ⇒ (4). It is clear, since R is an artinian ring, and R satisfies $(*)^{*}_{r}$.

(4) ⇒ (3). Let f ∈Pi(R) be any element. Using the argument as above, it follows that every E(fR) is uniform and projective. So that E(fR) ≅ eR for some e ∈Pi(R), by Lemma 3. Since fR is a waist in eR, then Z(eR) ≤ fR. It implies that Z(eR) = Z(fR). So that eR/Z(fR) has ACC on cyclic submodules, by (4). It follows that E(fR)/fR has ACC on cyclic submodules.

The proof of the theorem is complete. □

The main result in [5] is a special case of the following result.

Proposition 1

Let R be a ring satisfying the ACC (or DCC) on right annihilators. Then the following statements are equivalent:

• (1) R is a right co-Harada ring.

• (2) R is a right perfect ring and $R^{(2)}_{R}$ is an extending module.

• (3) R is a left perfect ring and $R^{(2)}_{R}$ is an extending module.

Proof

(1) ⇒ (2), (1) ⇒ (3) are obvious.

(2) ⇒ (1). Assume R is a right perfect ring. By [14], every right R-module has ACC on cyclic modules. By (2) ⇒ (1) in Theorem 3, it follows (1).

(3) ⇒ (1). Assume R is a left perfect ring. Let e ∈Pi(R). It easy to see that eR is a waist in E(eR) and eR/Z(eR) has finite length. It follows that R satisfies $(*)^{*}_{r}$, by (3) ⇒ (1) in Theorem 3, it follows (1). □

Corollary 1

Let R be a right nonsingular ring. Then the following statements are equivalent:

• (1) R is a right co-Harada ring.

• (2) R is a right perfect ring and $R^{(2)}_{R}$ is an extending module.

• (3) R is a left perfect ring and $R^{(2)}_{R}$ is an extending module.

• (4) R is Morita equivalent to a finite direct sum of upper triangular matrix rings over division rings.

Proof

It is shown by [3, Lemma 1.14] that if R is a ring with finite right Goldie then R has ACC and DCC on right annihilators. From Proposition 1 it easy to see the equivalence of (1),(2),(3). It follows from [16] that (1) ⇔ (4). □

It is also shown by Oshiro [16] that a ring R is a left Harada ring if and only if it is a right co-Harada ring, however a right Harada ring need not to be right co-Harada (see also Oshiro [16]).

Example 1

(Oshiro, [16]) There exists a right co-Harada ring which is not a right Harada ring. Consider the local QF ring Q = K[x,y]/(x²,y²), in which K is a field.

Put

$$ \begin{array}{@{}rcl@{}} J&=&J(Q),\quad S=\text{Soc}(Q_{Q}) \quad (=\text{Soc}(_{Q}Q)),\\ \bar{Q}&=&Q/S = \{\bar{a}~|~\bar{a}=a+S~\forall a \in Q \}. \end{array} $$

Defined V, W by

$$ V= \left[\begin{array}{ll} Q & Q \\ J & Q \end{array} \right],\quad W= \left[\begin{array}{ll} Q & \bar{Q} \\ J & \bar{Q} \end{array} \right]. $$

Then:

• (a) V is a Harada and co-Harada ring (right and left).

• (b) W is right co-Harada and left Harada. However, W is neither left co-Harada nor right Harada.

Finally we provide some characterizations of Harada rings (co-Harada rings) via perfect rings.

Theorem 4

Let R be a ring. Then the following statements are equivalent:

• (1) R is a Harada ring;

• (2) R is a co-Harada ring;

• (3) R is a perfect ring and $R^{(2)}_{R}$ and _RR⁽²⁾ are extending modules;

• (4) R is a perfect ring and every 2-generated right (or left) module M has a decomposition M = P ⊕ S₁ ⊕ S₂, where P is projective, S₁ is injective singular, S₂ is small singular;

• (5) R is a perfect ring, eR is a waist in E(eR) and Re is a waist in E(Re), for any e ∈Pi(R);

• (6) R is a perfect ring and M₂(R) is a CS ring.

Proof

(1) ⇔ (2). It follows from a result due to Oshiro [17] (see, Lemma 7). We shall prove that (2) ⇔ (4), (2) ⇒ (5) ⇒ (3) ⇒ (2) and (2) ⇔ (6).

(2) ⇒ (4). Assume R is a co-Harada ring. Let M be a 2-generated right R-module. Since R is a right co-Harada ring, M has a decomposition M = P ⊕ S, where P is projective and S is singular. Since R is a left co-H ring, it follows that R is a right Harada ring, by Lemma 7. Then, S has a decomposition S = S₁ ⊕ S₂, where S₁ is injective and S₂ is small. Hence M = P ⊕ S₁ ⊕ S₂, where P is projective, S₁ is injective and singular and S₂ is small and singular.

(4) ⇒ (2). Assume (4). Let M be any 2-generated right R-module. Then, M has a decomposition M = P ⊕ S₁ ⊕ S₂, where P is projective, S₁ is injective and singular and S₂ is small and singular. It means that M = P ⊕ (S₁ ⊕ S₂), where P is projective and S₁ ⊕ S₂ is singular. This shows that every 2-generated right R-module is a direct sum of a projective module and a singular module. By Lemma 5, it implies that $R^{(2)}_{R}$ is an extending module. Since R is perfect, it follows from [20, Proposition 3.4] that R satisfies the condition $(*)^{*}_{1,r}$. It is easy to see from Theorem 1 that every indecomposable projective right R-module is either injective or small. Let P = P₁ ⊕ P₂ ⊕⋯ ⊕ P_k. Let I = {1,2,…,k}. Let I₁ = {i ∈ I | P_i is injective} and I₂ = {j ∈ I | P_j is small}. It is easy to see that I = I₁ ∪ I₂ and I₁ ∩ I₂ = ∅. Whence, we have

$$ \begin{array}{@{}rcl@{}} M = P \oplus (S_{1} \oplus S_{2}) &= & \left( \oplus_{i \in I_{1}}P_{i}\right)\oplus \left( \oplus_{j \in I_{2}}P_{j}\right)\oplus (S_{1}\oplus S_{2}) \\ & = & \left( \left( \oplus_{i \in I_{1}}P_{i}\right) \oplus S_{1}\right) \oplus \left( \left( \oplus_{j \in I_{2}}P_{j}\right) \oplus S_{2}\right), \end{array} $$

where $(\oplus _{i \in I_{1}}P_{i} \oplus S_{1})$ is injective, and $(\oplus _{j \in I_{2}}P_{j} \oplus S_{2})$ is small by Lemma 6. Hence every 2-generated (and hence every cyclic) right R-module is a direct sum of an injective module and a small module. Next, we shall show that R satisfies the condition (∗)_r. Let X be a nonsmall right R-module. Since R is perfect, it follows from Lemma 6 that X contains a cyclic nonsmall module N. Then, N = N₁ + N₂, where N₁ is a nonzero injective module and N₂ is a small module. So, X contains nonzero injective module N₁. Therefore, R satisfies the condition (∗)_r. Since R is a perfect ring and R satisfies the condition (∗)_r, R is right artinian by [12]. Since R is right artinian and R satisfies the condition (∗)_r, R is a right Harada ring and so R is a left co-Harada ring. It remains to show that R is a right co-Harada ring. However this is clear since R is right artinian and R satisfies the condition $(*)^{*}_{1,r}$. Hence R is a co-Harada ring.

(2) ⇒ (5). Suppose that R is a co-Harada ring. Then R is an artinian ring (see [17, 18]). It follows from [20, Theorem 3.11] that eR is a waist in E(eR) and Re is a waist in E(Re).

(5) ⇒ (3). Assume (5). Since R is perfect and eR (resp. Re) is a waist in E(Re) (resp. E(Re)), it follows that $R^{(2)}_{R}$ (resp. _RR⁽²⁾) is an extending module, by [20, Theorem 2.4]. We have (3).

(3) ⇒ (2). Assume (3). If R is a left (resp. right) perfect ring and $R^{(2)}_{R}$ (resp. _RR⁽²⁾) is an extending module, the condition $(*)^{*}_{r}$ (resp. $(*)^{*}_{l}$) holds, by [20, Proposition 3.4]. Furthermore, R is a perfect (two-sided) QF-3⁺ ring, by [5]. Since R_R (resp. _RR) is isomorphic to a direct sum of modules of the form e_iR (resp. Re_i), we obtain a faithful injective right (resp. left) ideal by letting eR (resp. Re) be the sum of one of each isomorphism type of these e_iR (resp. Re_i). It implies that R is a right and left QF-3 ring. So, R is a semiprimary QF-3 ring. Hence R satisfies the ACC on right (and left) annihilators, by [4, Theorem 1.3]. Therefore R is a co-Harada ring.

(2) ⇒ (6). Since R is right and left co-Harada, then R is two-sided artinian and $R^{(2)}_{R}$, _RR⁽²⁾ are extending modules. By [7, Lemma 12.8], the ring M₂(R) is a right and left CS ring.

(6) ⇒ (2). Since M₂(R) is a right and left CS ring, then $R^{(2)}_{R}$ and _RR⁽²⁾ are extending modules, by [7, Lemma 12.8]. Since R is a perfect ring, we have (3); hence (2) holds.

The proof of the theorem is complete. □