Some Notes on Semiabelian Rings

Abstract

It is proved that if a ring 𝑅 is semiabelian, then so is the skew polynomial ring 𝑅[𝑥;𝜎], where 𝜎 is an endomorphism of 𝑅 satisfying 𝜎(𝑒)=𝑒 for all 𝑒∈𝐸(𝑅). Some characterizations and properties of semiabelian rings are studied.

1. Introduction

Throughout the paper, all rings are associative with identities. We always use 𝑁(𝑅) and 𝐸(𝑅) to denote the set of all nilpotent elements and the set of all idempotent elements of 𝑅.

According to [1], a ring 𝑅 is called semiabelian if every idempotent of 𝑅 is either right semicentral or left semicentral. Clearly, a ring 𝑅 is semiabelian if and only if either 𝑒𝑅(1−𝑒)=0 or (1−𝑒)𝑅𝑒=0 for every 𝑒∈𝐸(𝑅), so, abelian rings (i.e., every idempotent of 𝑅 is central) are semiabelian. But the converse is not true by [1, Example 2.2].

A ring 𝑅 is called directly finite if 𝑎𝑏=1 implies 𝑏𝑎=1 for any 𝑎,𝑏∈𝑅. It is well known that abelian rings are directly finite. In Theorem 2.7, we show that semiabelian rings are directly finite.

An element 𝑒 of a ring 𝑅 is called a left minimal idempotent if 𝑒∈𝐸(𝑅) and 𝑅𝑒 is a minimal left ideal of 𝑅. A ring 𝑅 is called left min-abel [2] if every left minimal idempotent element of 𝑅 is left semicentral. Clearly, abelian rings are left min-abel. In Theorem 2.7, we show that semiabelian rings are left min-abel.

A ring 𝑅 is called left idempotent reflexive if for any 𝑒∈𝐸(𝑅) and 𝑎∈𝑅, 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0. Theorem 2.5 shows that 𝑅 is abelian if and only if 𝑅 is semiabelian and left idempotent reflexive.

In [3], Wang called an element 𝑒 of a ring 𝑅 an op-idempotent if 𝑒²=−𝑒. Clearly, op-idempotent need not be idempotent. For example, let 𝑅=𝑍/3𝑍. Then 2∈𝑅 is an op-idempotent, while it is not an idempotent. In [4], Chen called an element 𝑒∈𝑅 potent in case there exists some integer 𝑛≥2 such that 𝑒^𝑛=𝑒. We write 𝑝(𝑒) for the smallest number 𝑛 of such. Clearly, idempotent is potent, while there exists a potent element which is not idempotent. For example, ¹⁰₀₋₁∈𝑀₂(𝑍) is a potent element, while it is not idempotent. We use 𝐸^o(𝑅) and 𝑃𝐸(𝑅) to denote the set of all op-idempotent elements and the set of all potent elements of 𝑅. In Corollaries 2.2 and 2.3, we observe that every semiabelian ring can be characterized by its op-idempotent and potent elements.

If 𝑅 is a ring and 𝜎∶𝑅→𝑅 is a ring endomorphism, let 𝑅[𝑥;𝜎] denote the ring of skew polynomials over 𝑅; that is all formal polynomials in 𝑥 with coefficients from 𝑅 with multiplication defined by 𝑥𝑟=𝜎(𝑟)𝑥. In [1], Chen showed that 𝑅 is a semiabelian ring if and only if 𝑅[𝑥] is a semiabelian ring. In Theorem 2.13, we show that if 𝑅 is a semiabelian ring with an endomorphism 𝜎 satisfying 𝜎(𝑒)=𝑒 for all 𝑒∈𝐸(𝑅), then 𝑅[𝑥;𝜎] is semiabelian.

2. Main Results

It is well known that an idempotent 𝑒 of a ring 𝑅 is left semicentral if and only if 1−𝑒 is right semicentral. Hence we have the following theorem.

Theorem 2.1. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is a semiabelian ring.(2)For any 𝑒∈𝐸(𝑅), 𝑒𝑅(1−𝑒)∪(1−𝑒)𝑅𝑒 is an ideal of 𝑅. (3)For any 𝑒∈𝐸(𝑅), 𝑒𝑅(1−𝑒)∪(1−𝑒)𝑅𝑒=𝑒𝑅(1−𝑒)+(1−𝑒)𝑅𝑒.

Proof. (1)⇒(2) assume that 𝑒∈𝐸(𝑅). Since 𝑅 is semiabelian, 𝑒 is either left semicentral or right semicentral. If 𝑒 is right semicentral, then 𝑒𝑅(1−𝑒)=0 and 1−𝑒 is left semicentral. Thus 𝑅(1−𝑒)𝑅𝑒𝑅=(1−𝑒)𝑅(1−𝑒)𝑅𝑒𝑅𝑒=(1−𝑒)𝑅𝑒 and 𝑒𝑅(1−𝑒)∪(1−v𝑒)𝑅𝑒=(1−𝑒)𝑅𝑒=𝑅(1−𝑒)𝑅𝑒𝑅 is an ideal of 𝑅. Similarly, if 𝑒 is left semicentral, then 𝑒𝑅(1−𝑒)∪(1−𝑒)𝑅𝑒=𝑅𝑒𝑅(1−𝑒)𝑅 is also an ideal of 𝑅.
(2)⇒(3) is clear.
(3)⇒(1) assume that 𝑒∈𝐸(𝑅). If 𝑒 is neither left semicentral nor right semicentral, there exist 𝑎,𝑏∈𝑅 such that (1−𝑒)𝑎𝑒≠0 and 𝑒𝑏(1−𝑒)≠0. By (3), (1−𝑒)𝑎𝑒+𝑒𝑏(1−𝑒)∈(1−𝑒)𝑅𝑒∪𝑒𝑅(1−𝑒). If (1−𝑒)𝑎𝑒+𝑒𝑏(1−𝑒)∈(1−𝑒)𝑅𝑒, then 𝑒𝑏(1−𝑒)=𝑒((1−𝑒)𝑎𝑒+𝑒𝑏(1−𝑒))(1−𝑒)∈𝑒((1−𝑒)𝑅𝑒)(1−𝑒)=0, a contradiction; if (1−𝑒)𝑎𝑒+𝑒𝑏(1−𝑒)∈𝑒𝑅(1−𝑒), then (1−𝑒)𝑎𝑒=0, it is also a contradiction. Hence 𝑒 is either left semicentral or right semicentral.

Evidently, 𝑅 is semiabelian if and only if either 𝑒𝑅(1−𝑒)=0 or (1−𝑒)𝑅𝑒=0 for every 𝑒∈𝐸(𝑅). On the other hand, an element 𝑒 of 𝑅 is op-idempotent if and only if −𝑒 is idempotent. Hence, by Theorem 2.1, we have the following corollary.

Corollary 2.2. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is a semiabelian ring.(2)For any 𝑒∈𝐸^𝑜(𝑅), 𝑒𝑅(1+𝑒)=0 or (1+𝑒)𝑅𝑒=0.(3)For any 𝑒∈𝐸^𝑜(𝑅), 𝑒𝑅(1+𝑒)∪(1+𝑒)𝑅𝑒 is an ideal of 𝑅.(4)For any 𝑒∈𝐸^𝑜(𝑅), 𝑒𝑅(1+𝑒)∪(1+𝑒)𝑅𝑒=𝑒𝑅(1+𝑒)+(1+𝑒)𝑅𝑒.

Clearly, for any 𝑒∈𝑃𝐸(𝑅), 𝑒^{𝑝(𝑒)−1}∈𝐸(𝑅), 𝑅𝑒=𝑅𝑒^{𝑝(𝑒)−1}, and 𝑒𝑅=𝑒^{𝑝(𝑒)−1}𝑅. Hence, by Theorem 2.1, we have the following corollary.

Corollary 2.3. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is a semiabelian ring.(2)For any 𝑒∈𝑃𝐸(𝑅), 𝑒𝑅(1−𝑒^{𝑝(𝑒)−1})=0 or (1−𝑒^{𝑝(𝑒)−1})𝑅𝑒=0.(3)For any 𝑒∈𝑃𝐸(𝑅), 𝑒𝑅(1−𝑒^{𝑝(𝑒)−1})∪(1−𝑒^{𝑝(𝑒)−1})𝑅𝑒 is an ideal of 𝑅.(4)For any 𝑒∈𝑃𝐸(𝑅), 𝑒𝑅(1−𝑒^{𝑝(𝑒)−1})∪(1−𝑒^{𝑝(𝑒)−1})𝑅𝑒=𝑒𝑅(1−𝑒^{𝑝(𝑒)−1})+(1−𝑒^{𝑝(𝑒)−1})𝑅𝑒.

Using Theorem 2.1, Corollaries 2.2 and 2.3, we have the following corollary.

Corollary 2.4. Let 𝑅 be a semiabelian ring. If 𝑒∈𝐸(𝑅), 𝑔∈𝐸^o(𝑅) and ℎ∈𝑃𝐸(𝑅), then:(1)if 𝑅𝑒𝑅=𝑅, then 𝑒=1,(2)if 𝑅𝑔𝑅=𝑅, then 𝑔=−1,(3)if 𝑅ℎ𝑅=𝑅, then ℎ^{𝑝(ℎ)−1}=1.

Call a ring 𝑅  idempotent reversible if 𝑔𝑅𝑒=0 implies 𝑒𝑅𝑔=0 for 𝑒,𝑔∈𝐸(𝑅). Clearly, abelian rings are left idempotent reflexive, and left idempotent reflexive rings are idempotent reversible. But we do not know that whether idempotent reversible rings must be left idempotent reflexive. It is easy to see that a ring 𝑅 is left idempotent reflexive if and only if for any 𝑎∈𝑁(𝑅), 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0. (In fact, it is only to show the sufficiency: Let 𝑎∈𝑅 and 𝑒∈𝐸(𝑅) satisfy 𝑎𝑅𝑒=0. If 𝑒𝑅𝑎=0, then 𝑒𝑏𝑎≠0 for some 𝑏∈𝑅. Since 𝑒𝑏𝑎∈𝑁(𝑅) and (𝑒𝑏𝑎)𝑅𝑒=0, by hypothesis, 𝑒𝑅(𝑒𝑏𝑎)=0, this implies 𝑒𝑏𝑎=𝑒𝑒(𝑒𝑏𝑎)=0, which is a contradiction. Hence 𝑒𝑅𝑎=0, 𝑅 is a left idempotent reflexive ring.)

Let 𝐷 be a division ring. Then the 2-by-2 upper triangular matrix ring UT₂(𝐷)=^𝐷𝐷_0𝐷 is not idempotent reversible. In fact, ¹⁰₀₀,⁰¹₀₁∈𝐸(UT₂(𝐷)) and ⁰¹₀₁^𝐷𝐷_0𝐷¹⁰₀₀=0, but ¹⁰₀₀^𝐷𝐷_0𝐷⁰¹₀₁=^0𝐷₀₀≠0. On the other hand, by [1, Example 2.2], UT₂(𝐷) is a semiabelian ring.

We have the following theorem.

Theorem 2.5. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is an abelian ring.(2)𝑅 is a semiabelian ring and idempotent reversible ring.(3)𝑅 is a semiabelian ring and left idempotent reflexive ring.(4)𝑅 is a semiabelian ring and for any 𝑎∈𝐽(𝑅), 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0.

Proof. (1)⇒(3)⇒(2)⇒(1) and (3)⇒(4) are trivial.
Now let 𝑒∈𝐸(𝑅). If 𝑅 is semiabelian, then 𝑒 is either right semicentral or left semicentral. If 𝑒 is right semicentral, then (1−𝑒)𝑅𝑒𝑅(1−𝑒)=0. Since 𝑅(1−𝑒)𝑅𝑒⊆𝑁(𝑅)∩𝐽(𝑅), (4) implies (1−𝑒)𝑅(1−𝑒)𝑅𝑒=0. Hence (1−𝑒)𝑅𝑒=0. This shows that 𝑒 is central; if 𝑒 is left semicentral, then 1−𝑒 is right semicentral. Hence 1−𝑒 and so 𝑒 is also central. Thus (4)⇒(1) holds.

Since semiprime rings are left idempotent reflexive, we have the following corollary by Theorem 2.5.

Corollary 2.6. Semiprime semiabelian rings are abelian.

Theorem 2.7. Let 𝑅 be a semiabelian ring and 𝑒∈𝐸(𝑅). Then,(1)𝑒𝑅(1−𝑒)𝑅𝑒=(1−𝑒)𝑅𝑒R(1−𝑒)=0,(2)If 𝑎∈𝑅 and 𝑎𝑒=0, then 𝑅𝑒𝑟𝑎⊆𝑁(𝑅) for all 𝑟∈𝑅,(3)𝑒𝑅(1−𝑒)⊆𝐽(𝑅).

Proof. (1) Since 𝑒 is right semicentral if and only if 𝑒𝑅(1−𝑒)=0 and 𝑒 is left semicentral if and only if (1−𝑒)𝑅𝑒=0, (1) is evident by hypothesis.(2)Since 𝑎=𝑎(1−𝑒), (𝑅𝑒𝑅𝑎)²=𝑅𝑒𝑅𝑎𝑅𝑒𝑅𝑎=𝑅𝑒𝑅𝑎(1−𝑒)𝑅𝑒𝑅𝑎=0 by (1). Hence 𝑅𝑒𝑅𝑎⊆𝑁(𝑅), so for any 𝑟∈𝑅, 𝑅𝑒𝑟𝑎⊆𝑁(𝑅).(3)Since (1−𝑒)𝑒=0, by (2), 𝑅𝑒𝑟(1−𝑒)⊆𝑁(𝑅) for all 𝑟∈𝑅. This implies 𝑅𝑒𝑟(1−𝑒)⊆𝐽(𝑅) for all 𝑟∈𝑅. Hence 𝑒𝑅(1−𝑒)⊆𝐽(𝑅).

Theorem 2.8. Let 𝑅 be a semiabelian ring. Then,(1)𝑅 is directly finite,(2)𝑅 is left min-abel.

Proof. (1) Assume that 𝑎𝑏=1. Let 𝑒=𝑏𝑎. Then 𝑒∈𝐸(𝑅) and 𝑒𝑏=𝑏. By Theorem 2.7(3), 𝑏(1−𝑒)=𝑒𝑏(1−𝑒)∈𝐽(𝑅). Hence 1−𝑒=𝑎𝑏(1−𝑒)∈𝐽(𝑅), which implies 1=𝑒=𝑏𝑎.
(2) Let 0≠𝑒∈𝐸(𝑅) and 𝑅𝑒 be a minimal left ideal of 𝑅. Then (1−𝑒)𝑅𝑒≠0 and 𝑅(1−𝑒)𝑅𝑒=𝑅𝑒. Since 𝑅 is a semiabelian ring, by Theorem 2.7(3), (1−𝑒)𝑅𝑒⊆𝐽(𝑅). This implies 𝑒∈𝐽(𝑅), that is, 𝑒=0 which is a contradiction. Hence (1−𝑒)𝑅𝑒=0, so 𝑒 is left semicentral. Hence 𝑅 is a left min-abel ring.

For a ring 𝑅, a proper left ideal 𝑃 of 𝑅 is prime if 𝑎𝑅𝑏⊆𝑃 implies that 𝑎∈𝑃 or 𝑏∈𝑃. Let Spec_𝑙(𝑅) be the set of all prime left ideals of 𝑅. In [5], it has been shown that if 𝑅 is not a left quasiduo ring, then Spec_𝑙(𝑅) is a space with the weakly Zariski topology but not with the Zariski topology.

Let 𝑅 be a ring. Then the set Max_𝑙(𝑅) of all maximal left ideals of 𝑅 is a compact 𝑇₁-space by [6, Lemma 2.1]. Recall that a topological space is said to be zero dimensional if it has a base consisting of clopen sets. Where a clopen set in a topological space is a set which is both open and closed.

Now, for a left ideal 𝐼 of a ring 𝑅, let 𝛼(𝐼)={𝑃∈Spec_𝑙(𝑅)∣𝐼⫅𝑃} and 𝛽(𝐼)=Spec_𝑙(𝑅)⧵𝛼(𝐼). If 𝐼=𝑅𝑎 for some 𝑎∈𝑅, then we write 𝛼_𝑙(𝑎) and 𝛽_𝑙(𝑎) for 𝛼(𝑅𝑎) and 𝛽(𝑅𝑎).

For any left ideal 𝐼 of 𝑅, we let 𝑈_𝑙(𝐼)=Max_𝑙(𝑅)∩𝛼(𝐼), 𝑉_𝑙(𝐼)=Max_𝑙(𝑅)∩𝛽(𝐼) and let 𝜉={𝑈_𝑙(∑_{1≤𝑖≤𝑛}∑_{1≤𝑗1<𝑗2<⋯<𝑗𝑖≤𝑛}(−1)^𝑖−1𝑒_𝑗1𝑒_𝑗1⋯𝑒_𝑗𝑖)∣𝑒_𝑗𝑖∈𝐸(𝑅),𝑖=1,2,…,𝑛,𝑛∈𝑍⁺}.

A ring 𝑅 is called left topologically boolean, or a left 𝑡𝑏-ring [7] for short, if for every pair of distinct maximal left ideals of 𝑅 there is an idempotent in exactly one of them.

A ring 𝑅 is called clean [8] if every element of 𝑅 is the sum of a unit and an idempotent.

The following theorems generalize [6, Lemmas 2.2 and 2.3].

Lemma 2.9. Let 𝑅 be a semiabelian ring and 𝑒_𝑖,𝑒,𝑓∈𝐸(𝑅),𝑖=1,2,…,𝑛. Then,(1)if 𝑁 is a maximal left ideal of 𝑅 and 𝑒∉𝑁, then 1−𝑒∈𝑁,(2)𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓)=𝑈_𝑙(𝑓𝑒),(3)𝑈_𝑙(𝑒)∪𝑈_𝑙(𝑓)=𝑈_𝑙(𝑒+𝑓−𝑒𝑓)=𝑈_𝑙(𝑒𝑓),(4)𝑈_𝑙(𝑒)=𝑉_𝑙(1−𝑒),(5)⋂^𝑛_𝑖=1𝑈_𝑙(𝑒_𝑖)=𝑈_𝑙(𝑒₁𝑒₂⋯𝑒_𝑛),(6)⋃^𝑛_𝑖=1𝑈_𝑙(𝑒_𝑖)=𝑈_𝑙(∑_{1≤𝑖≤𝑛}∑_{1≤𝑗1<𝑗2<⋯<𝑗𝑖≤𝑛}(−1)^𝑖−1𝑒_𝑗1𝑒_𝑗1…𝑒_𝑗𝑖),(7)𝑈_𝑙(∑_{1≤𝑖≤𝑛}∑_{1≤𝑗1<𝑗2<⋯<𝑗𝑖≤𝑛}(−1)^𝑖−1𝑒_𝑗1𝑒_𝑗1⋯𝑒_𝑗𝑖)=𝑉_𝑙(1−∑_{1≤𝑖≤𝑛}∑_{1≤𝑗1<𝑗2<⋯<𝑗𝑖≤𝑛}(−1)^𝑖−1𝑒_𝑗1𝑒_𝑗1…𝑒_𝑗𝑖).In particular, every set in 𝜉 is clopen.

Proof. (1) Since 𝑒∉𝑁, 𝑅𝑒+𝑁=𝑅. Let 1=𝑏𝑒+𝑛 for some 𝑏∈𝑅 and 𝑛∈𝑁. Since 𝑒𝑅(1−𝑒)𝑅𝑒=0 by Theorem 2.7(1), 𝑒𝑅(1−𝑒)=𝑒𝑅(1−𝑒)𝑛⊆𝑁. Since 𝑁 is a prime left ideal and 𝑒∉𝑁, 1−𝑒∈𝑁.
(2) Let 𝑃∈𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓). Then 𝑒∉𝑃 and 𝑓∉𝑃. By (1), we have 1−𝑒,1−𝑓∈𝑃. Hence 1−𝑒−𝑓+𝑒𝑓=(1−𝑒)(1−𝑓)∈𝑃. Clearly, 𝑒𝑓∉𝑃, so 𝑃∈𝑈_𝑙(𝑒𝑓). This shows 𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓)⊆𝑈_𝑙(𝑒𝑓). Conversely, if 𝑄∈𝑈_𝑙(𝑒𝑓), then 𝑒𝑓∉𝑄. Since 𝑄 is a left ideal, 𝑓∉𝑄. Hence 1−𝑓∈𝑄 by (1). If 𝑒∈𝑄, then 1−𝑒−𝑓+𝑒𝑓=(1−𝑒)(1−𝑓)∈𝑄 implies 𝑒𝑓∈𝑄, which is a contradiction. Hence 𝑒∉𝑄, so 𝑄∈𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓). Therefore 𝑈_𝑙(𝑒𝑓)⊆𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓). Thus 𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓)=𝑈_𝑙(𝑒𝑓). Similarly, we have 𝑈_𝑙(𝑒)∩𝑈_𝑙(𝑓)=𝑈_𝑙(𝑓𝑒).
(3) and (4) They are also straightforward to prove.
By induction on 𝑛, we can show (5), (6) and (7).
Thus every set in 𝜉 is clopen.

Theorem 2.10. Let 𝑅 be a semiabelian clean ring. Then 𝑅 is a left 𝑡𝑏-ring.

Proof. Suppose that 𝑀 and 𝑁 are distinct maximal left ideals of 𝑅. Let 𝑎∈𝑀⧵𝑁. Then 𝑅𝑎+𝑁=𝑅 and 1−𝑥𝑎∈𝑁 for some 𝑥∈𝑅. Clearly, 𝑥𝑎∈𝑀⧵𝑁. Since 𝑅 is clean, there exist an idempotent 𝑒∈𝐸(𝑅) and a unit 𝑢 in 𝑅 such that 𝑥𝑎=𝑒+𝑢. If 𝑒∈𝑀, then 𝑢=𝑥𝑎−𝑒∈𝑀 from which it follows that 𝑅=𝑀, a contradiction. Thus 𝑒∉𝑀. If 𝑒∉𝑁, then 1−𝑒∈𝑁 by Lemma 2.9 (1) and hence 𝑢=(1−𝑒)+(𝑥𝑎−1)∈𝑁. It follows that 𝑁=𝑅 which is also not possible. We thus have that 𝑒 is an idempotent belonging to 𝑁 only.

Theorem 2.11. Let 𝑅 be a semiabelian ring. If 𝑅 is a left 𝑡𝑏-ring, then 𝜉 forms a base for the weakly Zariski topology on Max_𝑙(𝑅). In particular, Max_𝑙(𝑅) is a compact, zero-dimensional Hausdorff space.

Proof. Similar to the proof of [6, Proposition 2.5], we can complete the proof.

A ring 𝑅 is called von Neumann regular if 𝑎∈𝑎𝑅𝑎 for all 𝑎∈𝑅 and 𝑅 is said to be unit-regular if for any 𝑎∈𝑅, 𝑎=𝑎𝑢𝑎 for some 𝑢∈𝑈(𝑅). A ring 𝑅 is called strongly regular if 𝑎∈𝑎²𝑅 for all 𝑎∈𝑅. Clearly, strongly regular ⇒ unit-regular ⇒von Neumann regular. Since von Neumann regular rings are semiprime, it follows that von Neumann regular rings are left idempotent reflexive. And it is well known that 𝑅 is strongly regular if and only if 𝑅 is von Neumann regular and abelian. In view of Theorem 2.5, we have the following corollary.

Corollary 2.12. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is strongly regular.(2)𝑅 is unit-regular and semiabelian.(3)𝑅 is von Neumann regular and semiabelian.

Following [9], a ring 𝑅 is called left 𝑁𝑃𝑃 if for any 𝑎∈𝑁(𝑅), 𝑅𝑎 is projective left 𝑅-module, and 𝑅 is said to be 𝑛-regular if for any 𝑎∈𝑁(𝑅), 𝑎∈𝑎𝑅𝑎. A ring 𝑅 is said to be reduced if 𝑎²=0 implies 𝑎=0 for each 𝑎∈𝑅, or equivalently, 𝑁(𝑅)=0. Obviously, reduced rings are 𝑛-regular and abelian, and 𝑛-regular rings are left 𝑁𝑃𝑃 and semiprime. Using Theorem 2.5, the following theorem gives some new characterization of reduced rings in terms of semiabelian rings.

Theorem 2.13. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is reduced.(2)𝑅 is 𝑛-regular and semiabelian.(3)𝑅 is left 𝑁𝑃𝑃, semiprime, and semiabelian.

Proof. (1)⇒(2)⇒(3) are trivial.
(3)⇒(1) let 𝑎∈𝑅 such that 𝑎²=0. Since 𝑅 is left 𝑁𝑃𝑃, 𝑙(𝑎)=𝑅𝑒,𝑒∈𝐸(𝑅). Hence 𝑒𝑎=0 and 𝑎=𝑎𝑒 because 𝑎∈𝑙(𝑎). Since 𝑅 is semiabelian and 𝑎𝑅𝑎=(1−𝑒)𝑎𝑒𝑅(1−𝑒)𝑎⊆(1−𝑒)𝑅𝑒𝑅(1−𝑒)𝑎, 𝑎𝑅𝑎=0 by Theorem 2.7. Since 𝑅 is semiprime, 𝑎=0, which shows that 𝑅 is reduced.

If 𝑅 is a ring and 𝜎∶𝑅→𝑅 is a ring endomorphism, let 𝑅[𝑥;𝜎] denote the ring of skew polynomials over 𝑅; that is all formal polynomials in 𝑥 with coefficients from 𝑅 with multiplication defined by 𝑥𝑟=𝜎(𝑟)𝑥. Note that if 𝑅(𝜎) is the (𝑅,𝑅)-bimodule defined by _𝑅𝑅(𝜎)=_𝑅𝑅 and 𝑚∘𝑟=𝑚𝜎(𝑟), for all 𝑚∈𝑅(𝜎) and 𝑟∈𝑅, then 𝑅[𝑥;𝜎]/(𝑥²)≅𝑅∝𝑅(𝜎).

Theorem 2.14. Let 𝑅 be a semiabelian ring. If 𝜎 is a ring endomorphism of 𝑅 satisfying 𝜎(𝑒)=𝑒 for all 𝑒∈𝐸(𝑅). Then 𝑅[𝑥;𝜎] is semiabelian.

Proof. Let 𝑓(𝑥)=𝑒₀+𝑒₁𝑥+⋯+𝑒_𝑛𝑥^𝑛∈𝐸(𝑅[𝑥;𝜎]). Then 𝑒²₀=𝑒₀,𝑒₁=𝑒₀𝑒₁+𝑒₁𝜎𝑒₀,𝑒₂=𝑒₀𝑒₂+𝑒₁𝜎𝑒₁+𝑒₂𝜎²𝑒₀,⋮𝑒_𝑛=𝑒₀𝑒_𝑛+𝑒₁𝜎𝑒_𝑛−1+𝑒₂𝜎²𝑒_𝑛−2+⋯+𝑒_𝑛−1𝜎^𝑛−1𝑒₁+𝑒_𝑛𝜎^𝑛𝑒₀. (2.1)
Since 𝑒₀∈𝐸(𝑅), 𝜎(𝑒₀)=𝑒₀ by hypothesis. Hence we have the following equations: 𝑒₁=𝑒₀𝑒₁+𝑒₁𝑒₀,𝑒₂=𝑒₀𝑒₂+𝑒₁𝜎𝑒₁+𝑒₂𝑒₀,⋮𝑒_𝑛=𝑒₀𝑒_𝑛+𝑒₁𝜎𝑒_𝑛−1+𝑒₂𝜎²𝑒_𝑛−2+⋯+𝑒_𝑛−1𝜎^𝑛−1𝑒₁+𝑒_𝑛𝑒₀. (2.2)
If 𝑒₀ is right semicentral, then 𝑒₀𝑒₁=𝑒₀𝑒₁+𝑒₀𝑒₁𝑒₀=𝑒₀𝑒₁+𝑒₀𝑒₁, which implies 𝑒₀𝑒₁=0. Hence 𝑒₁=𝑒₁𝑒₀.
Assume that 𝑒₀𝑒_𝑖=0 and 𝑒_𝑖=𝑒_𝑖𝑒₀ for 𝑖=1,2,…,𝑛−1. Then 𝑒₀𝑒_𝑛=𝑒₀𝑒_𝑛+𝑒₀𝑒_𝑛𝑒₀=𝑒₀𝑒_𝑛+𝑒₀𝑒_𝑛, (2.3) so 𝑒₀𝑒_𝑛𝑒=0,_𝑛=𝑒₁𝜎𝑒_𝑛−1+𝑒₂𝜎²𝑒_𝑛−2+⋯+𝑒_𝑛−1𝜎^𝑛−1𝑒₁+𝑒_𝑛𝑒₀=𝑒₁𝑒₀𝜎𝑒_𝑛−1+𝑒₂𝑒₀𝜎²𝑒_𝑛−2+⋯+𝑒_𝑛−1𝑒₀𝜎^𝑛−1𝑒₁+𝑒_𝑛𝑒₀=𝑒₁𝜎𝑒₀𝑒_𝑛−1+𝑒₂𝜎²𝑒₀𝑒_𝑛−2+⋯+𝑒_𝑛−1𝜎^𝑛−1𝑒₀𝑒₁+𝑒_𝑛𝑒₀=𝑒_𝑛𝑒₀. (2.4)
Hence 𝑓(𝑥)𝑒₀=𝑒₀+𝑒₁𝜎(𝑒₀)𝑥+⋯+𝑒_𝑛𝜎^𝑛(𝑒₀)𝑥^𝑛=𝑒₀+𝑒₁𝑥+⋯+𝑒_𝑛𝑥^𝑛=𝑓(𝑥) and 𝑒₀𝑓(𝑥)=𝑒₀.
For any 𝑔(𝑥)=𝑏₀+𝑏₁𝑥+⋯+𝑏_𝑚𝑥^𝑚∈𝑅[𝑥;𝜎], we have 𝑒₀𝑔(𝑥)𝑒₀=∑_{0≤𝑖≤𝑚}𝑒₀𝑏_𝑖𝜎^𝑖(𝑒₀)𝑥^𝑖=∑_{0≤𝑖≤𝑚}𝑒₀𝑏_𝑖𝑒₀𝑥^𝑖=∑_{0≤𝑖≤𝑚}𝑒₀𝑏_𝑖𝑥^𝑖=𝑒₀𝑔(𝑥). Thus 𝑓(𝑥)𝑔(𝑥)𝑓(𝑥)=𝑓(𝑥)𝑒₀𝑔(𝑥)𝑓(𝑥)=𝑓(𝑥)𝑒₀𝑔(𝑥)𝑒₀𝑓(𝑥)=𝑓(𝑥)𝑒₀𝑔(𝑥)𝑒₀=𝑓(𝑥)𝑒₀𝑔(𝑥)=𝑓(𝑥)𝑔(𝑥), which implies 𝑓(𝑥) is right semicentral in 𝑅[𝑥;𝜎]. Similarly, if 𝑒₀ is left semicentral in 𝑅, then we can show that 𝑓(𝑥) is left semicentral in 𝑅[𝑥;𝜎]. Hence 𝑅[𝑥;𝜎] is a semiabelian ring.

Corollary 2.15. Let 𝑅 be a semiabelian ring. If 𝜎 is a ring endomorphism of 𝑅 satisfying 𝜎(𝑒)=𝑒 for all 𝑒∈𝐸(𝑅). Then 𝑅[𝑥;𝜎]/(𝑥²) is semiabelian.

Proof. Since every element 𝑓(𝑥) of 𝑅[𝑥;𝜎]/(𝑥²) can be written 𝑓(𝑥)=𝑎₀+𝑎₁𝑥 with 𝑥²=0, by the same proof as Theorem 2.14, we can complete the proof.

Corollary 2.16. Let 𝑅 be a semiabelian ring. If 𝜎 is a ring endomorphism of 𝑅 satisfying 𝜎(𝑒)=𝑒 for all 𝑒∈𝐸(𝑅). Then 𝑅∝𝑅(𝜎) is semiabelian.

Corollary 2.17. Let 𝐷 be a division ring with an endomorphism 𝜎, then 𝐷[𝑥;𝜎]/(𝑥²) is semiabelian.

A ring 𝑅 is called left 𝑀𝐶2 [2] if for any 𝑎∈𝑅 and 𝑒∈𝑀𝐸_𝑙(𝑅), 𝑎𝑅𝑒=0 implies 𝑒𝑅𝑎=0. Clearly, left idempotent reflexive rings are left 𝑀𝐶2. We do not know whether idempotent reversible rings are left 𝑀𝐶2. But we know that there exists a left 𝑀𝐶2 ring which is not idempotent reversible. In fact, there exists a semiabelian ring 𝑅 which is not abelian (see the example above Theorem 2.5), by [1, Corollary 2.4], 𝑅[𝑥] is a semiabelian ring which is not abelian. Hence, by Theorem 2.5, 𝑅[𝑥] is not idempotent reversible. But 𝑅[𝑥] is a left 𝑀𝐶2 ring.

The authors in [10, Theorem 4.1] showed that if 𝑅 is a left 𝑀𝐶2 ring containing an injective maximal left ideal, then 𝑅 is a left self-injective ring. And [11, Proposition 5] showed that if 𝑅 is a left idempotent reflexive ring containing an injective maximal left ideal, then 𝑅 is a left self-injective ring.

Proposition 2.18. Let 𝑅 be an idempotent reversible 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 we have 𝑒𝑅(1−𝑒)=0. Since 𝑅 is idempotent reversible, (1−𝑒)𝑅𝑒=0. So 𝑒 is central. Now let 𝐿 be any proper essential left ideal of 𝑅 and 𝑓∶𝐿→𝑁 any non-zero 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)=𝑓(𝑥(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 [10, Proposition 5], we have that _𝑅𝑁 is injective. Hence 𝑅=𝑀⊕𝑁 is left self-injective.

Recall that a ring 𝑅 is left 𝑝𝑝 if every principal left ideal of 𝑅 is projective. As an application of Proposition 2.18, we have the following result.

Corollary 2.19. 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 an idempotent reversible left 𝑝𝑝 ring containing an injective maximal left ideal.

Proof. (1)⇒(2) is trivial.
(2)⇒(1) by Proposition 2.18, 𝑅 is a left self-injective ring. Hence, by [12, Theorem 1.2], 𝑅 is left 𝐶2, so, 𝑅 is von Neumann regular because 𝑅 is left 𝑝𝑝. Also we have Soc(_𝑅𝑅)≠0 since there is an injective maximal left ideal.

By [13], a ring 𝑅 is said to be left 𝐻𝐼 if 𝑅 is left hereditary containing an injective maximal left ideal. Osofsky [14] proved that left self-injective left hereditary ring is semisimple Artinian. We can generalize the result as follows.

Corollary 2.20. The following conditions are equivalent for a ring 𝑅.(1)𝑅 is a semisimple Artinian ring.(2)𝑅 is an idempotent reversible left 𝐻𝐼 ring.

According to [8], an element 𝑥∈𝑅 is called exchange if there exists 𝑒∈𝐸(𝑅) such that 𝑒∈𝑥𝑅 and 1−𝑒∈(1−𝑥)𝑅, and 𝑥 is said to be clean if 𝑥=𝑒+𝑢 where 𝑒∈𝐸(𝑅) and 𝑢∈𝑈(𝑅). By [8], clean elements are exchange and the converse holds when 𝑅 is an abelian ring. A ring 𝑅 is called exchange (clean) ring if every element of 𝑅 is an exchange (clean) element.

Proposition 2.21. Let 𝑅 be a semiabelian ring. If 𝑥∈𝑅 is an exchange element, then 𝑥 is a clean element.

Proof. Since 𝑥 is an exchange element, there exists 𝑒∈𝐸(𝑅) such that 𝑒∈𝑥𝑅 and 1−𝑒∈(1−𝑥)𝑅. Let 𝑒=𝑥𝑦 and 1−𝑒=(1−𝑥)𝑧 where 𝑦=𝑦𝑒,𝑧=𝑧(1−𝑒)∈𝑅. Then (𝑥−(1−𝑒))(𝑦−𝑧)=𝑥𝑦−𝑥𝑧−(1−𝑒)𝑦+(1−𝑒)𝑧=𝑥𝑦+(1−𝑥)𝑧−(1−𝑒)𝑦−𝑒𝑧=𝑒+1−𝑒−(1−𝑒)𝑦−𝑒𝑧=1−(1−𝑒)𝑦−𝑒𝑧. Since 𝑅 is a semiabelian ring, 𝑒 is either left semicentral or right semicentral. If 𝑒 is left semicentral, then (1−𝑒)𝑦=(1−𝑒)𝑦𝑒=0 and (𝑒𝑧𝑅)²=𝑒𝑧𝑅𝑒𝑧𝑅=𝑒𝑧(1−𝑒)𝑅𝑒𝑧𝑅⊆𝑒𝑅(1−𝑒)𝑅𝑒𝑅=0 by Theorem 2.7(1). Hence 𝑒𝑧∈𝐽(𝑅). Similarly, if 𝑒 is right semicentral, then 𝑒𝑧=𝑒𝑧(1−𝑒)=0 and (1−𝑒)𝑦∈𝐽(𝑅). This implies 1−(1−𝑒)𝑦−𝑒𝑧∈𝑈(𝑅), so (𝑥−(1−𝑒))(𝑦−𝑧)∈𝑈(𝑅). Since 𝑅 is a semiabelian ring, by Theorem 2.8, 𝑅 is a directly finite ring. Hence 𝑥−(1−𝑒)∈𝑈(𝑅), which implies 𝑥 is a clean element.

Corollary 2.22. If 𝑅 is a semiabelian exchange ring, then 𝑅 is a clean ring.

Theorem 2.23. Let 𝑅 be a semiabelian ring and 𝑎,𝑏∈𝑅. If 𝑎𝑏=0, then 𝑎𝐸(𝑅)𝑏⊆𝐽(𝑅).

Proof. Let 𝑎𝑏=0 and 𝑒∈𝐸(𝑅). Since 𝑅 is a semiabelian ring, either 𝑒 is left semicentral or 𝑒 is right semicentral. If 𝑒 is left semicentral, then (𝑅𝑎𝑒𝑏)²=𝑅𝑎𝑏𝑅𝑎𝑒𝑏=0. If 𝑒 is right semicentral, then (𝑅𝑎𝑒𝑏)²=𝑅𝑎𝑒𝑏𝑅𝑎𝑏=0. Hence 𝑅𝑎𝑒𝑏⊆𝐽(𝑅) for each 𝑒∈𝐸(𝑅), which implies 𝑎𝐸(𝑅)𝑏⊆𝐽(𝑅).

Corollary 2.24. Let 𝑅 be an abelian ring and 𝑎,𝑏∈𝑅. If 𝑎𝑏=0, then 𝑎𝐸(𝑅)𝑏⊆𝐽(𝑅).

The converse of Corollary 2.24 is not true, in general.

Example 2.25. let 𝐹 be a field, and 𝑅=^𝐹𝐹_0𝐹. Evidently, 𝐸(𝑅)=⋃_𝑥∈𝐹{⁰⁰₀₀,¹⁰₀₁,^1𝑥₀₀,^0𝑥₀₁}, 𝐽(𝑅)=^0𝐹₀₀. Let 𝐴=^𝑎1𝑏1_0𝑐1,𝐵=^𝑎2𝑏2_0𝑐2∈𝑅 and 𝐴𝐵=0. Then 𝑎₁𝑎₂=𝑐₁𝑐₂=0. Since 𝐴^1𝑥₀₀𝐵=^{0𝑎1𝑏2+𝑎1𝑥𝑐2}₀₀∈𝐽(𝑅) and 𝐴^0𝑥₀₁𝐵=^{0𝑎1𝑥𝑐2+𝑏1𝑐2}₀₀∈𝐽(𝑅). Hence 𝐴𝐸(𝑅)𝐵⊆𝐽(𝑅), but 𝑅 is not an abelian ring.

A ring 𝑅 is called EIFP if 𝑎,𝑏∈𝑅, 𝑎𝑏=0 implies 𝑎𝐸(𝑅)𝑏⊆𝐽(𝑅). Clearly, semiabelian rings are EIFP by Theorem 2.23. But the converse of Theorem 2.23 is not true, in general.

Example 2.26. Take the ring 𝑅 in Example 2.25, and let 𝑆=𝑅⊕𝑅. Then 𝑆 is EIFP, but not semiabelian. Indeed, take 𝑒₁=𝐸₁₁+𝐸₁₂ and 𝑒₂=𝐸₁₂+𝐸₂₂ in 𝑅, where 𝐸_𝑖𝑗 are matrix units. Then (𝑒₁;𝑒₂) is an idempotent. By a direct computation, (𝑒₁;𝑒₂) is neither left nor right semicentral. Hence 𝑅⊕𝑅 is not semiabelian while 𝑅⊕𝑅 is EIFP.

Acknowledgments

This paper was supported by the Foundation of Natural Science of China (10771182) and the Scientific Research Foundation of Graduate School of Jiangsu Province (CX09B_309Z). The authors would like to thank the referee for his/her helpful suggestions and comments.