Left 𝑊𝑀𝐶2 Rings

Abstract

We introduce in this paper the concept of left 𝑊𝑀𝐶2 rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left 𝐻𝐼 rings can be extended to left 𝑊𝑀𝐶2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with nonzero socle and extend some known results on strongly regular rings.

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Throughout this paper, 𝑅 denotes an associative ring with identity, and all modules are unitary. For any nonempty subset 𝑋 of a ring 𝑅, 𝑟(𝑋)=𝑟_𝑅(𝑋) and 𝑙(𝑋)=𝑙_𝑅(𝑋) denote the set of right annihilators of 𝑋 and the set of left annihilators of 𝑋, respectively. We use 𝐽(𝑅), 𝑁_∗(𝑅), 𝑁(𝑅), 𝑍_𝑙(𝑅), 𝐸(𝑅), 𝑆𝑜𝑐(_𝑅𝑅), and 𝑆𝑜𝑐(𝑅_𝑅) for the Jacobson radical, the prime radical, the set of all nilpotent elements, the left singular ideal, the set of all idempotent elements, the left socle, and the right socle of 𝑅, respectively.

An element 𝑘 of 𝑅 is called left minimal if 𝑅𝑘 is a minimal left ideal. An element 𝑒 of 𝑅 is called left minimal idempotent if 𝑒²=𝑒 is left minimal. We use 𝑀_𝑙(𝑅) and 𝑀𝐸_𝑙(𝑅) for the set of all left minimal elements and the set of all left minimal idempotent elements of 𝑅, respectively. Moreover, let 𝑀𝑃_𝑙(𝑅)={𝑘∈𝑀_𝑙(𝑅)∣_𝑅𝑅𝑘isprojective}.

A ring 𝑅 is called left 𝑀𝐶2 if every minimal left ideal which is isomorphic to a summand of _𝑅𝑅 is a summand. Left 𝑀𝐶2 rings were initiated by Nicholson and Yousif in [1]. In [2–6], the authors discussed the properties of left 𝑀𝐶2 rings. In [1], a ring 𝑅 is called left mininjective if 𝑟𝑙(𝑘)=𝑘𝑅 for every 𝑘∈𝑀_𝑙(𝑅), and 𝑅 is said to be left minsymmetric if 𝑘∈𝑀_𝑙(𝑅) always implies 𝑘∈𝑀_𝑟(𝑅). According to [1], left mininjective ⇒left minsymmetric ⇒left 𝑀𝐶2, and no reversal holds.

A ring 𝑅 is called left universally mininjective [1] if 𝑅𝑘 is an idempotent left ideal of 𝑅 for every 𝑘∈𝑀_𝑙(𝑅). The work in [2] uses the term left 𝐷𝑆 for the left universally mininjective. According to [1, Lemma  5.1], left 𝐷𝑆 rings are left mininjective.

A ring 𝑅 is called left min-abel [3] if for each 𝑒∈𝑀𝐸_𝑙(𝑅), 𝑒 is left semicentral in 𝑅, and 𝑅 is said to be strongly left min-abel [3, 7] if every element of 𝑀𝐸_𝑙(𝑅) is central in 𝑅.

A ring 𝑅 is called left 𝑊𝑀𝐶2 if 𝑔𝑅𝑒=0 implies 𝑒𝑅𝑔=0 for 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑔∈𝐸(𝑅).

Let 𝐹 be a field and 𝑅={^𝑎𝑏_0𝑎∣𝑎,𝑏∈𝐹}. Then 𝐸(𝑅)={⁰⁰₀₀,¹⁰₀₁} and 𝑀𝐸_𝑙(𝑅) is empty, so 𝑅 is left 𝑊𝑀𝐶2. Now let 𝑆=^𝐹𝐹_0𝐹. Then 𝑀𝐸_𝑙(𝑆)={^1𝑢₀₀∣𝑢∈𝐹} and 𝐸(𝑆)={⁰⁰₀₀,¹⁰₀₁,^1𝑢₀₀,^0𝑢₀₁∣𝑢∈𝐹}. Since ^0𝑢₀₁𝑆¹⁰₀₀=0 and ¹⁰₀₀𝑆^0𝑢₀₁≠0, 𝑆 is not left 𝑊𝑀𝐶2.

Let 𝑅 be any ring and 𝑆₁=𝑅[𝑥] and 𝑆₂=𝑅[[𝑥]]. Then 𝑀𝐸_𝑙(𝑆₁) and 𝑀𝐸_𝑙(𝑆₂) are all empties, so 𝑆₁ and 𝑆₂ are all left 𝑊𝑀𝐶2.

A ring 𝑅 is called left idempotent reflexive [8] if 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0 for all 𝑎∈𝑅 and 𝑒∈𝐸(𝑅). Clearly, 𝑅 is left idempotent reflexive if and only if for any 𝑎∈𝑁(𝑅) and 𝑒∈𝐸(𝑅), 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0 if and only if for any 𝑎∈𝐽(𝑅) and 𝑒∈𝐸(𝑅), 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0. Therefore, left idempotent reflexive rings are left 𝑊𝑀𝐶2.

In general, the existence of an injective maximal left ideal in a ring 𝑅 can not guarantee the left self-injectivity of 𝑅. In [9], Osofsky proves that if 𝑅 is a semiprime ring containing an injective maximal left ideal, then 𝑅 is left self-injective. In [8], Kim and Baik prove that if 𝑅 is left idempotent reflexive containing an injective maximal left ideal, then 𝑅 is left self-injective. In [10], Wei and Li prove that if 𝑅 is left 𝑀𝐶2 containing an injective maximal left ideal, then 𝑅 is left self-injective. Motivated by these results, in this paper, we show that if 𝑅 is a left 𝑊𝑀𝐶2 ring containing an injective maximal left ideal, then 𝑅 is left self-injective. As an application of this result, we show that a ring 𝑅 is a semisimple Artinian ring if and only if 𝑅 is a left 𝑊𝑀𝐶 ring and left 𝐻𝐼 ring.

We start with the following theorem.

Theorem 1. The following conditions are equivalent for a ring 𝑅: (1)𝑅 is left 𝑀𝐶2;(2)for any 𝑎∈𝑅 and 𝑒∈𝑀𝐸_𝑙(𝑅), 𝑒𝑎𝑅𝑒=0 implies 𝑒𝑎=0;(3)for any 𝑒,𝑔∈𝑀𝐸_𝑙(𝑅), (𝑔−𝑒)𝑅𝑒=0 implies that 𝑒=𝑒𝑔;(4)for any 𝑘,𝑙∈𝑀_𝑙(𝑅), 𝑘𝑅𝑙=0 implies 𝑙𝑅𝑘=0.

Proof. (1) ⇒(2) Assume that 𝑎∈𝑅 and 𝑒∈𝑀𝐸_𝑙(𝑅) with 𝑒𝑎𝑅𝑒=0. If 𝑒𝑎≠0, then 𝑅𝑒𝑎≅𝑅𝑒. By (1), 𝑅𝑒𝑎=𝑅𝑔 for some 𝑔∈𝑀𝐸_𝑙(𝑅). Hence 𝑅𝑔=𝑅𝑔𝑅𝑔=𝑅𝑒𝑎𝑅𝑒𝑎=𝑅(𝑒𝑎𝑅𝑒)𝑎=0, which is a contradiction. Hence 𝑒𝑎=0.
(2) ⇒(3) Let 𝑒,𝑔∈𝑀𝐸_𝑙(𝑅) such that (𝑔−𝑒)𝑅𝑒=0. Then 𝑒(𝑔−𝑒)𝑅𝑒=0. By (2), 𝑒(𝑔−𝑒)=0. Hence 𝑒=𝑒𝑔.
(3) ⇒(4) Assume that 𝑘,𝑙∈𝑀_𝑙(𝑅) with 𝑘𝑅𝑙=0. If 𝑙𝑅𝑘≠0, then 𝑅𝑙𝑅𝑘=𝑅𝑘. Hence 𝑅𝑘=𝑅𝑙𝑅𝑘=(𝑅𝑙)²𝑅𝑘, which implies 𝑅𝑙=𝑅𝑒 for some 𝑒∈𝑀𝐸_𝑙(𝑅). Since 𝑅𝑒𝑅𝑘=𝑅𝑙𝑅𝑘=𝑅𝑘≠0, there exists 𝑏∈𝑅 such that 𝑒𝑏𝑘≠0. Let 𝑔=𝑒+𝑒𝑏𝑘. Then 𝑔²=𝑒+𝑒𝑏𝑘+𝑒𝑏𝑘𝑒+𝑒𝑏𝑘𝑒𝑏𝑘=𝑒+𝑒𝑏𝑘=𝑔∈𝑀𝐸_𝑙(𝑅) because 𝑒𝑏𝑘𝑒∈𝑅𝑘𝑅𝑒=𝑅𝑘𝑅𝑙=0 and 𝑔≠0. Since (𝑔−𝑒)𝑅𝑒=𝑒𝑏𝑘𝑅𝑒=𝑒𝑏𝑘𝑅𝑙=0, by (3), 𝑒=𝑒𝑔. Hence 𝑔=𝑒𝑔=𝑒, which implies 𝑒𝑏𝑘=0. It is a contradiction. Therefore 𝑙𝑅𝑘=0.
(4) ⇒(1) Let 𝑎∈𝑀_𝑙(𝑅) and 𝑒∈𝑀𝐸_𝑙(𝑅) with 𝑅𝑎≅𝑅𝑒. Then there exists 𝑔∈𝑀𝐸_𝑙(𝑅) such that 𝑎=𝑔𝑎 and 𝑙(𝑎)=𝑙(𝑔). If (𝑅𝑎)²=0, then 𝑅𝑎𝑅⊆𝑙(𝑎)=𝑙(𝑔), so 𝑎𝑅𝑔=0, by (4), 𝑔𝑅𝑎=0, which implies 𝑎=𝑔𝑎=0. It is a contradiction. Hence (𝑅𝑎)²≠0, so 𝑅𝑎=𝑅ℎ for some ℎ∈𝑀𝐸_𝑙(𝑅), which implies 𝑅 is a left 𝑀𝐶2 ring.

Corollary 2. Left 𝑀𝐶2 rings are left 𝑊𝑀𝐶2.

Proof. Let 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑔∈𝐸(𝑅) with 𝑔𝑅𝑒=0. If 𝑒𝑅𝑔≠0, then 𝑒𝑏𝑔≠0 for some 𝑏∈𝑅. Clearly, 𝑒𝑏𝑔∈𝑀_𝑙(𝑅) and (𝑒𝑏𝑔)𝑅𝑒=0. Since 𝑅 is a left 𝑀𝐶2 ring, by Theorem 1, 𝑒𝑅(𝑒𝑏𝑔)=0, which implies 𝑒𝑏𝑔=0, and this is a contradiction. Hence 𝑒𝑅𝑔=0 and so 𝑅 is a left 𝑊𝑀𝐶2 ring.

We do not know whether the converse of Corollary 2 holds. However, we have the following characterization of left WMC2 rings.

Theorem 3. Let 𝑅 be a ring. Then the following conditions are equivalent: (1)𝑅 is a left 𝑊𝑀𝐶2 ring;(2)for any 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑔∈𝐸(𝑅), 𝑒𝑔≠0 implies 𝑔𝑅𝑒≠0;(3)for any 𝑒∈𝑀𝐸_𝑙(𝑅), 𝑙(𝑅𝑒)∩𝐸(𝑅)⊆𝑟(𝑒𝑅);(4)for any 𝑘∈𝑀𝑃_𝑙(𝑅) and 𝑔∈𝐸(𝑅), 𝑔𝑅𝑘=0 implies 𝑘𝑅𝑔=0.

Proof. (4) ⇒(1) ⇒(2) It is easy to show by the definition of left 𝑊𝑀𝐶2 ring.
(2) ⇒(3) Let 𝑔∈𝑙(𝑅𝑒)∩𝐸(𝑅). Then 𝑔𝑅𝑒=0. We claim that 𝑒𝑅𝑔=0. Otherwise, there exists 𝑏∈𝑅 such that 𝑒𝑏𝑔≠0. Clearly, ℎ=𝑒𝑏𝑔+𝑔−𝑒𝑔∈𝐸(𝑅) and 𝑒ℎ=𝑒𝑏𝑔≠0. By (2), we have ℎ𝑅𝑒≠0. But ℎ𝑅𝑒=0 because 𝑔𝑅𝑒=0. This is a contradiction. Hence 𝑒𝑅𝑔=0 and so 𝑔∈𝑟(𝑒𝑅). Therefore 𝑙(𝑅𝑒)∩𝐸(𝑅)⊆𝑟(𝑒𝑅).
(3) ⇒(4) Since 𝑘∈𝑀𝑃_𝑙(𝑅), _𝑅𝑅𝑘 is projective. It is easy to show that 𝑘=𝑒𝑘 and 𝑙(𝑘)=𝑙(𝑒) for some 𝑒∈𝑀𝐸_𝑙(𝑅). Since 𝑔𝑅𝑘=0, 𝑔𝑅⊆𝑙(𝑘). Therefore 𝑔𝑅𝑒=0, which implies 𝑔∈𝑙(𝑅𝑒)∩𝐸(𝑅). By (3), 𝑒𝑅𝑔=0. Hence 𝑘𝑅𝑔=𝑒𝑘𝑅𝑔⊆𝑒𝑅𝑔=0.

By Theorem 3, we have the following corollary.

Corollary 4. (1) Let 𝑅 be a left 𝑊𝑀𝐶2 ring. If 𝑒∈𝐸(𝑅) satisfying 𝑅𝑒𝑅=𝑅, then 𝑒𝑅𝑒 is left 𝑊𝑀𝐶2.
(2) If 𝑅 is a direct product of a family rings {𝑅_𝑖∶𝑖∈𝐼}, then 𝑅 is a left 𝑊𝑀𝐶2 ring if and only if every 𝑅_𝑖 is left 𝑊𝑀𝐶2.

Theorem 5. (1) If 𝑅 is a subdirect product of a family left 𝑊𝑀𝐶2 rings {𝑅_𝑖∶𝑖∈𝐼}, then 𝑅 is a left 𝑊𝑀𝐶2 ring.
(2) If 𝑅/𝑍_𝑙(𝑅) is a left 𝑊𝑀𝐶2 ring, so is 𝑅.

Proof. (1) Let 𝑅_𝑖=𝑅/𝐴_𝑖, where 𝐴_𝑖 are ideals of 𝑅 with ⋂_𝑖∈𝐼𝐴_𝑖=0. Let 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑔∈𝐸(𝑅) satisfying 𝑔𝑅𝑒=0. For any 𝑖∈𝐼, if 𝑒∈𝐴_𝑖, then 𝑒𝑅𝑔∈𝐴_𝑖; if 𝑒∉𝐴_𝑖, then we can easily show that 𝑒_𝑖=𝑒+𝐴_𝑖∈𝑀𝐸_𝑙(𝑅_𝑖). Since 𝑅_𝑖 is a left 𝑊𝑀𝐶2 ring and 𝑔_𝑖𝑅_𝑖𝑒_𝑖=0, where 𝑔_𝑖=𝑔+𝐴_𝑖, 𝑒_𝑖𝑅_𝑖𝑔_𝑖=0. Hence 𝑒𝑅𝑔⊆𝐴_𝑖. In any case, we have 𝑒𝑅𝑔⊆𝐴_𝑖 for all 𝑖∈𝐼. Therefore 𝑒𝑅𝑔⊆⋂_𝑖∈𝐼𝐴_𝑖=0 and so 𝑒𝑅𝑔=0. This shows that 𝑅 is a left 𝑊𝑀𝐶2 ring.
(2) Let 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑔∈𝐸(𝑅) satisfying 𝑒𝑔≠0. Clearly, in 𝑅=𝑅/𝑍_𝑙(𝑅), 𝑒=𝑒+𝑍_𝑙(𝑅)∈𝑀𝐸_𝑙(𝑅), 𝑔=𝑔+𝑍_𝑙(𝑅)∈𝐸(𝑅). Since _𝑅𝑅𝑒𝑔≅_𝑅𝑅𝑒, 𝑒𝑔∉𝑍_𝑙(𝑅). Since 𝑅 is a left 𝑊𝑀𝐶2 ring, by Theorem 3, 𝑔𝑅𝑒≠0, which implies 𝑔𝑅𝑒≠0. Thus 𝑅 is a left 𝑊𝑀𝐶2 ring by Theorem 3.

Theorem 6. (1)  𝑅 is a strongly left min-abel ring if and only if 𝑅 is a left min-abel left 𝑊𝑀𝐶2 ring.
(2) If 𝑅/𝑍_𝑙(𝑅) is a strongly left min-abel ring, then so is 𝑅.

Proof. (1) Theorem  1.8 in [3] shows that 𝑅 is a strongly left min-abel ring if and only if 𝑅 is a left min-abel left 𝑀𝐶2 ring, so by Corollary 2, we obtain that strongly left min-abel ring is left min-abel left 𝑊𝑀𝐶2.
Conversely, let 𝑅 be a left min-abel left 𝑊𝑀𝐶2 ring. Let 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑎∈𝑅 satisfying 𝑒𝑎𝑅𝑒=0. Set 𝑔=1−𝑒+𝑒𝑎. Then, clearly, 𝑔∈𝐸(𝑅) and 𝑒𝑔=𝑒𝑎. Since 𝑅 is a left min-abel ring, (1−𝑒)𝑅𝑒=(1−𝑒)𝑒𝑅𝑒=0, so 𝑔𝑅𝑒=0. Since 𝑅 is a left 𝑊𝑀𝐶2 ring, 𝑒𝑅𝑔=0, which implies 𝑒𝑎=𝑒𝑔=0, by Theorem 1, 𝑅 is a left 𝑀𝐶2 ring. Hence 𝑅 is a strongly left min-abel ring.
(2) It is an immediate corollary of (1), [3, Corollary  1.5(2)] and Theorem 5(2).

A ring 𝑅 is called left idempotent reflexive [8] if 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0 for all 𝑎∈𝑅 and 𝑒∈𝐸(𝑅). Clearly, left idempotent reflexive rings are left 𝑊𝑀𝐶2.

In general, the existence of an injective maximal left ideal in a ring 𝑅 cannot guarantee the left self-injectivity of 𝑅. Proposition  5 in [8] proves that if 𝑅 is a left idempotent reflexive ring containing an injective maximal left ideal, then 𝑅 is a left self-injective ring. Theorem  4.1 in [10] proves that if 𝑅 is a left MC2 ring containing an injective maximal left ideal, then 𝑅 is a left self-injective ring. We can generalize the results as follows.

Theorem 7. Let 𝑅 be a left 𝑊𝑀𝐶2 ring. If 𝑅 contains an injective maximal left ideal, then 𝑅 is a left self-injective ring.

Proof. Let 𝑀 be an injective maximal left ideal of 𝑅. Then 𝑅=𝑀⊕𝑁 for some minimal left ideal 𝑁 of 𝑅. Hence we have 𝑀=𝑅𝑒 and 𝑁=𝑅(1−𝑒) for some 𝑒²=𝑒∈𝑅. If 𝑀𝑁=0, then 𝑒𝑅(1−𝑒)=0. Since 𝑅 is left 𝑊𝑀𝐶2 and 1−𝑒∈𝑀𝐸_𝑙(𝑅), (1−𝑒)𝑅𝑒=0. So 𝑒 is central. Now let 𝐿 be any proper essential left ideal of 𝑅 and 𝑓∶𝐿→𝑁 any nonzero left 𝑅−homomorphism. Then 𝐿/𝑈≅𝑁, where 𝑈=ker𝑓 is a maximal submodule of 𝐿. Now 𝐿=𝑈⊕𝑉, where 𝑉≅𝑁=𝑅(1−𝑒) is a minimal left ideal of 𝑅. Since 𝑒 is central, 𝑉=𝑅(1−𝑒). For any 𝑧∈𝐿, let 𝑧=𝑥+𝑦, where 𝑥∈𝑈, 𝑦∈𝑉. Then 𝑓(𝑧)=𝑓(𝑥)+𝑓(𝑦)=𝑓(𝑦). Since 𝑦=𝑦(1−𝑒)=(1−𝑒)𝑦, 𝑓(𝑧)=𝑓(𝑦)=𝑓(𝑦(1−𝑒))=𝑦𝑓(1−𝑒). Since 𝑥(1−𝑒)=(1−𝑒)𝑥∈𝑉∩𝑈=0, 𝑥𝑓(1−𝑒)=𝑓𝑥(1−𝑒)=𝑓(0)=0. Thus 𝑓(𝑧)=𝑦𝑓(1−𝑒)=𝑦𝑓(1−𝑒)+𝑥𝑓(1−𝑒)=(𝑦+𝑥)𝑓(1−𝑒)=𝑧𝑓(1−𝑒). Hence _𝑅𝑁 is injective. If 𝑀𝑁≠0, by the proof of [11, Proposition  5], we have _𝑅𝑁 is injective. Hence 𝑅=𝑀⊕𝑁 is left self-injective.

A ring 𝑅 is called strongly left 𝐷𝑆 [3] if 𝑘²≠0 for all 𝑘∈𝑀_𝑙(𝑅). Since strongly left 𝐷𝑆 ⇒left 𝐷𝑆 ⇒left mininjective ⇒left minsymmetric ⇒left 𝑀𝐶2 ⇒left 𝑊𝑀𝐶2 and strongly left min-abel ⇒left 𝑊𝑀𝐶2, we have the following corollary.

Corollary 8. Let 𝑅 contain an injective maximal left ideal. If 𝑅 satisfies one of the following conditions, then 𝑅 is a left self-injective ring. (1)𝑅 is a strongly left 𝐷𝑆 ring.(2)𝑅 is a left 𝐷𝑆 ring.(3)𝑅 is a left mininjective ring.(4)𝑅 is a left minsymmetric ring.(5)𝑅 is a strongly left min-abel ring.(6)𝑅 is a left 𝑀𝐶2 ring.

It is well known that if 𝑅 is a left self-injective ring, then 𝐽(𝑅)=𝑍_𝑙(𝑅). Therefore by [2, Theorem  5.1] and Corollary 8, we have the following corollary.

Corollary 9. Let 𝑅 contain an injective maximal left ideal. Then 𝑅 is left self-injective if and only if 𝐽(𝑅)=𝑍_𝑙(𝑅).

A ring 𝑅 is called left 𝑛𝑖𝑙-injective [5] if for any 𝑎∈𝑁(𝑅), 𝑟𝑙(𝑎)=𝑎𝑅, and 𝑅 is said to be left 𝑁𝐶2 [5] if for any 𝑎∈𝑁(𝑅), _𝑅𝑅𝑎 is projective implies that 𝑅𝑎=𝑅𝑒 for some 𝑒∈𝐸(𝑅). By [5, Theorem  2.22], left 𝑛𝑖𝑙-injective rings are left 𝑁𝐶2 and left 𝑁𝐶2 rings are left 𝑀𝐶2. A ring 𝑅 is right Kasch if every simple right 𝑅-module can be embedded in 𝑅_𝑅, and 𝑅 is said to be left 𝐶2 [12] if every left ideal that is isomorphic to a direct summand of _𝑅𝑅 is itself a direct summand. Clearly, left self-injective rings are left 𝐶2 [13] and left 𝐶2 rings are left 𝑁𝐶2 and by [14, Lemma  1.15], right Kasch rings are left 𝐶2. Hence, we have the following corollary.

Corollary 10. (1) Let 𝑅 contain an injective maximal left ideal. Then the following conditions are equivalent: (a)𝑅 is a left self-injective ring;(b)𝑅 is a left nil-injective ring;(c)𝑅 is a left 𝐶2 ring;(d)𝑅 is a left 𝑁𝐶2 ring.
(2) If 𝑅 is a right Kasch ring containing an injective maximal left ideal, then 𝑅 is a left self-injective ring.

A ring 𝑅 is called left min-𝐴𝑃-injective if for any 𝑘∈𝑀_𝑙(𝑅), 𝑟𝑙(𝑘)=𝑘𝑅⊕𝑋_𝑘, where 𝑋_𝑘 is a right ideal of 𝑅. Clearly, left mininjective rings are left min 𝐴𝑃-injective.

Lemma 11. (1) If 𝑅 is a left min -𝐴𝑃-injective ring, then 𝑅 is left 𝑊𝑀𝐶2.
(2) If 𝑆𝑜𝑐(_𝑅𝑅)⊆𝑆𝑜𝑐(𝑅_𝑅), then 𝑅 is left 𝑊𝑀𝐶2.

Proof. (1) Let 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑔∈𝐸(𝑅) satisfying 𝑒𝑔≠0. Since 𝑅 is a left min-𝐴𝑃-injective ring and 𝑙(𝑒)=𝑙(𝑒𝑔), 𝑒𝑅=𝑟𝑙(𝑒)=𝑟𝑙(𝑒𝑔)=𝑒𝑔𝑅⊕𝑋_𝑒𝑔, where 𝑋_𝑒𝑔 is a right ideal 𝑅. Set 𝑒=𝑒𝑔𝑏+𝑥, 𝑏∈𝑅 and 𝑥∈𝑋_𝑒𝑔. Then 𝑒𝑔=𝑒(𝑒𝑔)=𝑒𝑔𝑏𝑒𝑔+𝑥𝑒𝑔, so 𝑥𝑒𝑔=𝑒𝑔−𝑒𝑔𝑏𝑒𝑔∈𝑒𝑔𝑅∩𝑋_𝑒𝑔, which implies 𝑥𝑒𝑔=0, so 𝑒𝑔=𝑒𝑔𝑏𝑒𝑔. Let ℎ=𝑒𝑔𝑏. Then ℎ∈𝑀𝐸_𝑙(𝑅) and 𝑒𝑔𝑅=ℎ𝑅. Therefore ℎ𝑅=ℎ𝑅ℎ𝑅=𝑒𝑔𝑅𝑒𝑔𝑅 which implies 𝑔𝑅𝑒≠0. By Theorem 3, 𝑅 is a left 𝑊𝑀𝐶2 ring.
(2) Assume that 𝑒∈𝑀𝐸_𝑙(𝑅) and 𝑎∈𝑅 satisfying 𝑒𝑎𝑅𝑒=0. If 𝑒𝑎≠0, then 𝑒𝑎∈𝑆𝑜𝑐(_𝑅𝑅)⊆𝑆𝑜𝑐(𝑅_𝑅). Thus there exists a minimal right ideal 𝑘𝑅 of 𝑅 such that 𝑘𝑅⊆𝑒𝑎𝑅. Clearly, 𝑙(𝑘)=𝑙(𝑒𝑎)=𝑙(𝑒) and 𝑘𝑅𝑘𝑅⊆𝑒𝑎𝑅𝑒𝑎𝑅=0. Hence 𝑅𝑘𝑅⊆𝑙(𝑘). Let 𝐼 be a complement right ideal of 𝑅𝑘𝑅 in 𝑅. Then 𝐼⊆𝑙(𝑘) and 𝑒∈𝑆𝑜𝑐(_𝑅𝑅)⊆𝑆𝑜𝑐(𝑅_𝑅)⊆𝑅𝑘𝑅⊕𝐼⊆𝑙(𝑘)=𝑙(𝑒), which is a contradiction. Hence 𝑒𝑎=0. By Theorem 1, 𝑅 is a left 𝑀𝐶2 ring, so 𝑅 is left 𝑊𝑀𝐶2 by Corollary 2.

Since left mininjective rings are left min-𝐴𝑃-injective and 𝑆𝑜𝑐(_𝑅𝑅)⊆𝑆𝑜𝑐(𝑅_𝑅). Hence by Theorem 7, Corollary 8 and Lemma 11, we have the following theorem.

Theorem 12. Let 𝑅 contain an injective maximal left ideal. Then the following conditions are equivalent: (1)𝑅 is left self-injective;(2)𝑅 is left min-𝐴𝑃-injective;(3)𝑆𝑜𝑐(_𝑅𝑅)⊆𝑆𝑜𝑐(𝑅_𝑅).

A ring 𝑅 is called(1)strongly reflexive if 𝑎𝑅𝑏𝑅𝑐=0 implies 𝑎𝑅𝑐𝑅𝑏=0 for all 𝑎,𝑏,𝑐∈𝑅;(2)reflexive [8, 15] if 𝑎𝑅𝑏=0 implies 𝑏𝑅𝑎=0 for all 𝑎,𝑏∈𝑅;(3)symmetric if 𝑎𝑏𝑐=0 implies 𝑎𝑐𝑏=0 for all 𝑎,𝑏,𝑐∈𝑅;(4)𝑍𝐶 [16] if 𝑎𝑏=0 implies 𝑏𝑎=0 for all 𝑎,𝑏∈𝑅;(5)𝑍𝐼 [16] if 𝑎𝑏=0 implies 𝑎𝑅𝑏=0 for all 𝑎,𝑏∈𝑅.

Evidently, we have the following proposition.

Proposition 13. (1) The following conditions are equivalent for a ring 𝑅: (a)𝑅 is semiprime;(b)𝑅 is strongly reflexive and every proper essential right ideal of 𝑅 contains no nonzero nilpotent ideal;(c)𝑅 is reflexive and every proper essential right ideal of 𝑅 contains no nonzero nilpotent ideal;(d)𝑅 is strongly reflexive and 𝑁_∗(𝑅)∩𝑍_𝑙(𝑅)=0;(e)𝑅 is reflexive and 𝑁_∗(𝑅)∩𝑍_𝑙(𝑅)=0.
(2)𝑅 is symmetric if and only if 𝑅 is 𝑍𝐼 and strongly reflexive.
(3)𝑅 is reversible if and only if 𝑅 is 𝑍𝐼 and reflexive.
(4) Strongly reflexive ⇒reflexive ⇒left idempotent reflexive.

It is well known that if 𝑅 is a left self-injective ring, then 𝑍_𝑙(𝑅)=𝐽(𝑅), so 𝑅/𝑍_𝑙(𝑅) is semiprimitive. Hence 𝑅/𝑍_𝑙(𝑅) is left 𝑊𝑀𝐶2 by Proposition 13. Thus, by Theorems 5 and 7, we have the following theorem.

Theorem 14. Let 𝑅 contain an injective maximal left ideal. Then (1)𝑅 is a left self-injective ring if and only if 𝑅/𝑍_𝑙(𝑅) is a left 𝑊𝑀𝐶2 ring.(2)If 𝑅 satisfies one of the following conditions, then 𝑅 is a left self-injective:(a)𝑅 is a semiprime ring;(b)𝑅 is a strongly reflexive;(c)𝑅 is reflexive;(d)𝑅 is a left idempotent reflexive.

Recall that a ring 𝑅 is left 𝑝𝑝 if every principal left ideal of 𝑅 is projective as left 𝑅-module. As an application of Theorem 7, we have the following result.

Theorem 15. The following conditions are equivalent for a ring 𝑅: (1)𝑅 is a von Neumann regular left self-injective ring with 𝑆𝑜𝑐(_𝑅𝑅)≠0;(2)𝑅 is a left 𝑊𝑀𝐶2 left 𝑝𝑝 ring containing an injective maximal left ideal;(3)𝑅 is a left 𝑝𝑝 ring containing an injective maximal left ideal and 𝑅/𝑍_𝑙(𝑅) is a left 𝑊𝑀𝐶2 ring.

Proof. (1) ⇒(3) is trivial.
(3) ⇒(2) is a direct result of Theorem 5(2).
(2) ⇒(1) By Theorem 7, 𝑅 is left self-injective. Hence, by [13, Theorem  1.2], 𝑅 is left 𝐶2, so 𝑅 is von Neumann regular because 𝑅 is left 𝑝𝑝. In addition, we have 𝑆𝑜𝑐(_𝑅𝑅)≠0 since there is an injective maximal left ideal.

By [17], a ring 𝑅 is said to be left 𝐻𝐼 if 𝑅 is left hereditary containing an injective maximal left ideal. Osofsky [9] proves that a left self-injective left hereditary ring is semisimple Artinian. We can generalize the result as follows.

Corollary 16. The following conditions are equivalent for a ring 𝑅: (1)𝑅 is semisimple Artinian;(2)𝑅 is left 𝑊𝑀𝐶2 left 𝐻𝐼;(3)𝑅 is left 𝑁𝐶2 left 𝐻𝐼;(4)𝑅 is left min-𝐴𝑃-injective left 𝐻𝐼;(5)𝑅 is left idempotent reflexive left 𝐻𝐼.

Acknowlegment

Project supported by the Foundation of Natural Science of China (10771182, 10771183).