Regularity of n-P-V-Rings and n-P-V’-Rings

Abstract

The regularity of the n-P-V-rings and n-P-V’-rings is systematically investigated in this paper. Employing the notions of quasi-ideals, weakly left (or right) ideals, and generalized weak ideals, we focus on investigating the strong $\pi$-regularity and weak $\pi$-regularity of the n-P-V-rings and the n-P-V’-rings. Subsequently, we demonstrate our results as follows: (1) $R$ is strongly $\pi$-regular if $R$ is a left n-P-V-ring where all its maximal left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals. (2) $R$ is strongly $\pi$-regular iff $R$ is an abelian left (right) n-P-V’-ring where all its maximal essential left (right) ideals are either quasi-ideals, weakly right (left) ideals, or generalized weak ideals. (3) $R$ is reduced left weakly $\pi$-regular if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are either quasi-ideals, weakly right ideals, or generalized weak ideals.

1. Introduction

Von Neumann first introduced the notion of regular rings in 1936 in the Proceedings of the National Academy of Sciences of the United States of America (PNAS). He defined a regular ring $R$ as one in which, for any element $x\in R$, there exists an element $c\in R$ such that $x=xcx$. Due to their desirable algebraic properties, investigating the regularity of von Neumann regular rings has become a significant area of study [1,2,3,4].

Since Eckmann and Schopf introduced the notion of injective modules in 1953, injectivity has become a powerful tool in ring and module theory [5]. Notably, Kaplansky demonstrated that a commutative ring $R$ is a von Neumann regular iff all its simple $R$-modules are injective (i.e., $R$ is defined as a V-ring) [6], establishing a strong connection between V-rings and the regular rings.

However, the focus in ring theory often shifts to noncommutative and associative rings. For such rings, the relationship between regularity and injectivity becomes more intricate. In 1973, Michler and Villamayor showed that the conditions relating to V-rings and von Neumann regular rings in the noncommutative setting are neither sufficient nor necessary [7]. This result underscores the complex and intriguing interplay between these two classes of rings in the noncommutative context.

As one of the generalizations of injective modules, Zhang and Wu presented the notion of P-injective modules in the literature [8] and used P-injective modules to investigate von Neumann regular rings, which naturally led to the notion of P-V-rings. A further study in the literature [9] demonstrates that the P-injective module is weakened, and the concept of a GP-injective module is obtained. As a result, the notion of V-rings has been generalized to include GP-V-rings and GP-V’-rings. Right (left) GP-V-rings, where every simple right (left) $R$-module is a GP-injective module, have been studied in [10], while right (left) GP-V’-rings, where any simple singular right (left) $R$-module is a GP-injective module, have been characterized in [11]. In recent years, the regularity of GP-V-rings and GP-V’-rings has attracted significant attention from many mathematicians (cf. [12,13]).

In 2004, Mao introduced the notion of n-P-injective modules and n-P-rings [14]. A left $R$-module is said to be left n-P-injective if there exists a positive integer $n$ such that, for any $c\in R$, any module homomorphism $f$ from $R{c}^n$ to $M$ can be extended to a module homomorphism from ${}_{R}R$ to $M$. It is evident that a left 1-P-injective module must be a type of left P-injective module. Any left P-injective module must also be a left n-P-injective module, and any left n-P-injective module must be a left GP-injective module; however, the converse statements do not necessarily hold. Thus, left n-P-injective modules lie between left P-injective modules and left GP-injective modules, representing a proper generalization of left P-injective modules.

In [15], Kim was the first to introduce the concept of idempotent reflexive rings. However, no research has been conducted on idempotent reflexive n-P-V’-rings and their regularity properties until now. This paper addresses this gap by systematically examining the regularity of idempotent reflexive n-P-V’-rings.

The organization of this paper is as follows. Section 2 introduces pertinent lemmas and definitions relevant to this study. Section 3 delves into the investigation of the strong $\pi$-regularity and weak $\pi$-regularity of the n-P-V-rings and the n-P-V’-rings. Section 4 gives the conclusions of the paper and prospects for future research.

2. Preliminaries

We begin by presenting several definitions and lemmas to provide a clearer foundation for the results that follow.

Definition 1

([15]). The left ideal $N$ of the ring $R$ is idempotent reflexive if $rRe\subseteq N$ implies $eRr\subseteq N$ for $everyr\in R$ and every idempotent $e$ of $R$. The ring $R$ is an idempotent reflexive ring if $rRe=0$ implies $eRr=0$ for $everyr\in R$ and every idempotent $e$ of $R$.

It is clear that semiprime rings and abelian rings all possess idempotent reflexivity. However, Kim [15] provided an example showing that a ring with idempotent reflexivity is not necessarily reflexive.

Definition 2.

The ring $R$ is said to be semi-commutative if $xRy=0$ for every $x,y\in R$ and $xy=0$.

Definition 3.

The ring $R$ is said to be semi-abelian iff either $ere=er$ or $ere=re$ for all $r\in R$ and every $e^2=e\in R$.

Definition 4.

The ring $R$ is said to be reduced if the ring $R$ contains no nonzero nilpotent elements.

Definition 5.

Let $R$ be a ring and $G$ be an additive semigroup of the ring $R$. If there is a positive integer $p$ satisfying ${{\left(rR\right)}^p\subseteq G(or\left(Rr\right)}^p\subseteq G)$ for all $r\in R$, then the additive semigroup $G$ is said to be a weakly right (left) ideal.

Definition 6.

Let $R$ be a ring and $N$ be a right (left) ideal of $R$. If there is a positive integer $m$ such that $R{r}^m\subseteq N(or{r}^{m}R\subseteq N)$ for all $r\in R$, then the right (left) ideal $N$ is called a generalized weak ideal.

Definition 7.

Let $R$ be a ring and $G$ be an additive semigroup of $R$. If $grg\in G,rgr\in G$ for $allg\in G,andr\in R$, then the additive semigroup $G$ is called a quasi-ideal.

Definition 8

([14]). The ring R is said to be a left (right) n-P-V-ring if there is a positive integer n such that any simple left (right) $R$-module is n-P-injective.

Definition 9.

The ring R is said to be a left (right) n-P-V’-ring if there is a positive integer n satisfying any simple singular left (right) $R$-module is n-P-injective.

Definition 10.

The ring R is said to be a $\pi$-regular ring if there is a positive integer $m$ and c $\in R$ satisfying $r^m=r^{m}c{r}^m$ for all $r\in R$.

In 2002, Tuganbaev [16] introduced the notion of strongly $\pi$-regular rings.

Definition 11.

The ring R is a strongly $\pi$-regular ring if there is a positive integer m and a $\in R$ satisfying $r^m=r^{m+1}a$ for all $r\in R$.

Clearly, when $m=1$, a $\pi$-regular ring is simply a regular ring, thus the $\pi$-regular ring is a generalization of the regular ring. Tuganbaev pointed out in the literature [16] that a strongly $\pi$-regular ring is unequivocally a $\pi$-regular ring, but a $\pi$-regular ring does not invariably qualify as strongly $\pi$-regular; consequently, a regular ring is not necessarily strongly $\pi$-regular.

Definition 12.

The ring R is a left weakly $\pi$-regular ring if there is a positive integer m satisfying $r^{m}R={\left(r^{m}R\right)}^2$ for all $r\in R$.

Since n-P-injective modules form a class of injective modules that lie between P-injective modules and GP-injective modules, it follows that, under similar conditions, n-P-V-rings and n-P-V’-rings typically exhibit better properties than GP-V-rings and GP-V’-rings. Based on this observation, we explore the regularity theorems related to GP-V’-rings as discussed in [13] and conclude that n-P-V-rings and n-P-V’-rings demonstrate a stronger regularity under the same conditions.

The subsequent discussion focuses on the regularity of n-P-V-rings and n-P-V’-rings. To derive the relevant theorems, we initially provide several preliminary lemmas.

Lemma 1.

$R$ is a reduced ring if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are quasi-ideals.

Proof. 

Let there be a nonzero element $c\in R$ satisfying $c^2=0$, then we can get $c\in l\left(c\right)$. Hence, there is a maximal left ideal $L$ of the ring $R$ containing $l\left(c\right)$, and it is evident that $L$ is an essential ideal. Indeed, if $L$ is not an essential ideal, $L$ is a direct sum term of the ring $R$ and $L=l\left(e\right)$, in which $e$ is an idempotent of the ring $R$ and $c\in l\left(c\right)\subseteq L=l\left(c\right),$ then $ce=0$. Given that the ring $R$ is semi-abelian, we deduce that $ec=0$, indicating $e\in l\left(c\right)\subseteq L=l\left(e\right).$ Hence, $e=e^2=0$ which contradicts the assumption. Thus $L$ is an essential ideal and $l\left(c\right)\subseteq L$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R,$$

(1)

and it is clear that $f$ qualifies as a well-defined left $R$-module homomorphism. By assumption, $R/L$ is a left n-P-injective module, so there is an element $a\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}a+L,1-c^{n}a\in L,$$

(2)

where $c^n\in l\left(c\right)\subseteq L$. Considering that $L$ is a left ideal and a quasi-ideal, we have

$$\left(1+a\right)c^n\left(1+a\right)\in L,a{c}^{n}a\in L,a{c}^n\in L.$$

(3)

Therefore, we derive

$$c^{n}a=\left(1+a\right)c^n\left(1+a\right)-c^n-a{c}^n-a{c}^{n}a\in L.$$

(4)

Thus, $1=\left(1-c^{n}a\right)+c^{n}a\in L$. This is in conflict with the maximality of $L$. Therefore, $R$ is a reduced ring. □

Lemma 2.

The ring $R$ is reduced if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are weakly right ideals.

Proof. 

Let there be a nonzero element $c\in R$ satisfying $c^2=0$, then we can get $c\in l\left(c\right)$. Hence, there is a maximal left ideal $L$ of the ring $R$ containing $l\left(c\right)$, and it is evident that $L$ is an essential ideal. Indeed, if $L$ is not an essential ideal, $L$ is a direct sum term of the ring $R$ and $L=l\left(e\right)$, in which $e$ is an idempotent of the ring $R$ and $c\in l\left(c\right)\subseteq L=l\left(c\right),$ then $ce=0$. Given that the ring $R$ is semi-abelian, we deduce that $ec=0$, indicating $e\in l\left(c\right)\subseteq L=l\left(e\right)$. Hence, $e=e^2=0$ which contradicts the assumption. Thus, $L$ is an essential ideal and $l\left(c\right)\subseteq L$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(5)

It is clear that $f$ qualifies as a well-defined left $R$-module homomorphism. By assumption, $R/L$ is a left n-P-injective module, so there is an element $a\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}a+L,1-c^{n}a\in L,$$

(6)

where $c^n\in l\left(c\right)\subseteq L$. Considering that $L$ is both a left ideal and a weakly right, there exists a positive integer $p$ satisfying $(c^{n}a{)}^p\in L$. Therefore, we derive

$$1-(c^{n}a{)}^p=\left[1+c^{n}a+(c^{n}a{)}^2+\cdots (c^{n}a{)}^{p-1}\right](1-c^{n}a)\in L.$$

(7)

Thus $1=\left[1-(c^{n}a{)}^p\right]+(c^{n}a{)}^p\in L$. This is in conflict with the maximality of $L$. Hence, the ring $R$ is reduced. □

Lemma 3.

The ring R is reduced if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are generalized weak ideals.

Proof. 

Let there be a nonzero element $c\in R$ satisfying $c^2=0$, then we can get $c\in l\left(c\right)$. Hence, there is a maximal left ideal $L$ of the ring $R$ containing $l\left(c\right)$, and it is evident that $L$ is an essential ideal. Indeed, if $L$ is not an essential ideal, $L$ is a direct sum term of the ring $R$ and $L=l\left(e\right)$, in which $e$ is an idempotent of the ring $R$ and $c\in l\left(c\right)\subseteq L=l\left(c\right),$ then $ce=0$. Given that the ring $R$ is semi-abelian, we deduce that $ec=0$, indicating $e\in l\left(c\right)\subseteq L=l\left(e\right)$. Hence, $e=e^2=0$ which contradicts the assumption. Thus $L$ is an essential ideal and $l\left(c\right)\subseteq L$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(8)

It is clear that $f$ qualifies as a well-defined left $R$-module homomorphism. By assumption, $R/L$ is a left n-P-injective module, so there is an element $a\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}a+L,1-c^{n}a\in L,$$

(9)

where $c^n\in l\left(c\right)\subseteq L$. Considering that $L$ is a left ideal, we have $a{c}^n\in L, a(1-c^{n}a)=a-{ac}^{n}a\in L$. In consideration of $L$ is a generalized weak ideal, there is a positive integer $m$ satisfying $(a{c}^n{)}^{m}a\in L$. Therefore, we can derive

$$\begin{gathered} (a{c}^n{)}^{m-1}a=(a{c}^n{)}^{m-1}(a-a{c}^{n}a)+(a{c}^n{)}^{m}a\in L, \\ (a{c}^n{)}^{m-2}a=(a{c}^n{)}^{m-2}(a-a{c}^{n}a)+(a{c}^n{)}^{m-1}a\in L, \\ \dots \dots \dots \dots \dots \dots \dots \\ \begin{array}{c}a{c}^{n}a=a{c}^n(a-a{c}^{n}a)+(a{c}^n{)}^{2}a\in L,\\ a=(a-a{c}^{n}a)+a{c}^{n}a\in L.\end{array} \end{gathered}$$

(10)

Given that $L$ is a left ideal of the ring $R$, we have $c^{n}a\in L$. Thus, $1=\left(1-c^{n}a\right)+c^{n}a\in L$. This is in conflict with the maximality of $L$. Hence, $R$ is a reduced ring. □

3. Main Results

We first investigated the regularity of n-P-V-rings. Under the condition that “all its maximal left ideals are generalized weak ideals”, we proved that the left n-P-V-ring is strongly $\pi$-regular.

Theorem 1.

$R$ is a strongly $\pi$-regular ring if $R$ is a left n-P-V-ring where all its maximal left ideals are generalized weak ideals.

Proof. 

Suppose there is an element c $\in R$ satisfying $Rc+l\left(c^n\right)\ne R$. Then the ring $R$ contains a maximal left ideal $L$ satisfying $Rc+l\left(c^n\right)\subseteq L\subset R$. Thus, $R/L$ is a simple left $R$-module. Given that every maximal left ideal of the ring $R$ is a generalized weak ideal and $c^n\in L$, we have $b{c}^n\in L$ for all $b\in R$. We define the following left $R$-module homomorphism as:

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(11)

Based on the assumption, $R/L$ is a left n-P-injective module, so there is an element $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L,$$

(12)

and it is obvious that

$$1-c^{n}b\in L,b\left(1-c^{n}b\right)=b-b{c}^{n}b\in L.$$

(13)

Since $b{c}^n\in L$, and $L$ is a generalized weak ideal of the ring $R$, then there is a positive integer $m$ satisfying $(b{c}^n{)}^{m}b\in L$. Thus, we obtain

$$(b{c}^n{)}^{m-1}b=(b{c}^n{)}^{m-1}(b-b{c}^{n}b)+(b{c}^n{)}^{m}b\in L.$$

(14)

Continuing this process, we can obtain $b{c}^{n}b\in L$. Since $b=\left(b-b{c}^{n}b\right)+b{c}^{n}b\in L$, this is in conflict with the maximality of $L$. Therefore, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Thus, there exists $a\in R$ and $d\in l\left(c^n\right)$ such that

$$ac+d=1,$$

(15)

$$a{c}^{n+1}=c^n.$$

(16)

Hence, the ring $R$ is strongly $\pi$-regular. □

The substitution of “generalized weak ideals” with “weakly right ideals” does not alter the conclusion of Theorem 1.

Theorem 2.

$R$ is a strongly $\pi$-regular ring if $R$ is a left n-P-V-ring where all its maximal left ideals are weakly right ideals.

Proof. 

Suppose there is an element c $\in R$ satisfying $Rc+l\left(c^n\right)\ne R$. Then the ring $R$ contains a maximal left ideal $L$ such that $Rc+l\left(c^n\right)\subseteq L\subset R$. Therefore, $R/L$ is a simple left $R$-module. Since all maximal left ideals of the ring $R$ are weakly right ideals and $c^n\in L$, we have $b{c}^n\in L$ for all $b\in R$. We define the following left $R$-module homomorphism as:

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(17)

Based on this assumption, $R/L$ is a left n-P-injective module, so there is an element $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L.$$

(18)

It is obvious $1-c^{n}b\in L$. Since $L$ is a weakly right ideal and a left ideal, there is a positive integer $m$ satisfying $(c^{n}b{)}^m\in L$. Therefore, we can derive

$$\begin{gathered} 1-\left(c^{n}b\right)\in L, \\ {c}^{n}b(1-c^{n}b)=c^{n}b-(c^{n}b{)}^2\in L, \\ \left(c^{n}b{)}^2\right(1-c^{n}b)=(c^{n}b{)}^2-(c^{n}b{)}^3\in L, \\ \begin{array}{c}\dots \dots \dots \dots \end{array} \\ \left(c^{n}b{)}^{m-1}\right(1-c^{n}b)=(c^{n}b{)}^{m-1}-(c^{n}b{)}^m\in L. \end{gathered}$$

(19)

Thus, we obtain

$$1-(c^{n}b{)}^m\in L,1\in L.$$

(20)

which is in conflict with the maximality of $L$. Therefore, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Thus, there exists $a\in R$ and $d\in l\left(c^n\right)$ such that

$$ac+d=1,$$

(21)

$$a{c}^{n+1}=c^n.$$

(22)

Hence, the ring $R$ is strongly $\pi$-regular. □

The substitution of “weakly right ideals” with “quasi-ideals” does not alter the conclusion of Theorem 2.

Theorem 3.

$R$ is a strongly $\pi$-regular ring if $R$ is a left n-P-V-ring in which all its maximal left ideals are quasi-ideals.

Proof. 

Suppose there is an element c $\in R$ satisfying $Rc+l\left(c^n\right)\ne R$. Then the ring $R$ contains a maximal left ideal such that $Rc+l\left(c^n\right)\subseteq L\subset R$. Thus, $R/L$ is a simple left $R$-module. Since all maximal left ideals of the ring $R$ are quasi-ideals and $c^n\in L$, we have $b{c}^n\in L$ for all $b\in R$. We define the following left $R$-module homomorphism as:

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(23)

Based on this assumption, $R/L$ is a left n-P-injective module, so there is an element $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L,$$

(24)

and it is obvious that $1-c^{n}b\in L$. Since $c^n\in l\left(c\right)\subseteq L$, and $L$ is a quasi-ideal and a left ideal, we derive

$$\left(1+b\right)c^n\left(1+b\right)\in L,b{c}^n\in L.$$

(25)

Hence,

$$c^{n}b=\left(1+b\right)c^n\left(1+b\right)-b{c}^n-b{c}^{n}b-c^n\in L,$$

(26)

and thus, we obtain $1=\left(1-c^{n}b\right)+c^{n}b\in L$. This is in conflict with the maximality of the ideal $L$. Therefore, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Thus, there exists $a\in R$ and $d\in l\left(c^n\right)$ such that

$$ac+d=1,$$

(27)

$$a{c}^{n+1}=c^n.$$

(28)

Hence, $R$ is strongly $\pi$-regular. □

In the following, we start to study the regularity of the n-P-V’-rings, and we first present three equivalent conditions for the n-P-V’-rings to be strongly $\pi$-regular.

Theorem 4.

The following statements are equivalent:

(1) 

$R$ is a strongly $\pi$-regular ring.

(2) 

$R$ is an abelian left n-P-V’-ring in which all its maximal essential left ideals are quasi-ideals.

(3) 

$R$ is an abelian right n-P-V’-ring in which all its maximal essential right ideals are quasi-ideals.

Proof. 

(1) ⇒ (2) and (1) ⇒ (3) are trivial. Since (3) ⇒ (1) is similar to (2) ⇒ (1), we only need (2) ⇒ (1) in the following.

For (2) ⇒ (1): Suppose there is an element c $\in R$ satisfying $Rc+l\left(c^n\right)\ne R$. Then the ring $R$ contains a maximal left ideal $L$ such that

$$Rc+l\left(c^n\right)\subseteq L\subset R.$$

(29)

In the following, we prove that $L$ is essential. Otherwise, $L$ is a direct sum term of the ring $R$, i.e., $L=l\left(e\right)$, where $e\ne 0, \mathrm{and} e^2=e\in R.$ Then $Rc+l\left(c^n\right)\subseteq L=l\left(e\right)$, so we have

$$Rce=0.$$

(30)

Thus, $ce=0$. Noting that $R$ is abelian, so we obtain

$$ec=ce=0.$$

(31)

Hence,

$$e\in l\left(c\right)\subseteq l\left(c^n\right)\subseteq L=l\left(e\right).$$

(32)

Thus, we obtain

$$e=e^2=0,$$

(33)

which contradicts the hypothesis. Therefore, $L$ is an essential ideal of the ring $R$, and $R/L$ is a simple singular left $R$-module. Since all maximal essential left ideals of $R$ are quasi-ideals, and $c^n\in L$, we have $b{c}^n\in L$ for all $b\in R$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(34)

It is clear that $f$ qualifies as a well-defined left $R$-module homomorphism. By assumption, $R/L$ is an n-P-injective module, so there is $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L,$$

(35)

and it is obvious that $1-c^{n}b\in L$. By the fact that $L$ is the left ideal, we have $b\left(1-c^{n}b\right)\in L$. Since $c^n\in l\left(c\right)\subseteq L$, and $L$ is the left ideal, we obtain $b{c}^n\in L$. Given that $L$ is a quasi-ideal, we derive

$$b{c}^{n}b\in L,\left(1+b\right)c^n\left(1+b\right)\in L.$$

(36)

Thus,

$$c^{n}b=\left(1+b\right)c^n\left(1+b\right)-b{c}^n-c^n-b{c}^{n}b\in L.$$

(37)

Hence,

$$1=\left(1-c^{n}b\right)+c^{n}b\in L.$$

(38)

This is in conflict with the maximality of the ideal $L$. Thus, we obtain $Rc+l\left(c^n\right)=R$ for all $c\in R$. Therefore, $R$ is a strong $\pi$-regular ring. □

Theorem 5.

The following statements are equivalent:

(1) 

$R$ is a strongly $\pi$-regular ring.

(2) 

$R$ is an abelian left n-P-V’-ring in which all its maximal essential left ideals are weakly right ideals.

(3) 

$R$ is an abelian right n-P-V’-ring in which all its maximal essential right ideals are weakly left ideals.

Proof. 

(1) ⇒ (2) and (1) ⇒ (3) are trivial. Since (3) ⇒ (1) is similar to (2) ⇒ (1), we only need (2) ⇒ (1) in the following.

For (2) ⇒ (1): Suppose there is an element c $\in R$ satisfying $Rc+l\left(c^n\right)\ne R$. Then the ring $R$ contains a maximal left ideal $L$ such that

$$Rc+l\left(c^n\right)\subseteq L\subset R.$$

(39)

In the following, we prove that $L$ is essential. Otherwise, $L$ is a direct sum term of the ring $R$, i.e., $L=l\left(e\right)$, where $e\ne 0, \mathrm{and} e^2=e\in R.$ Then $Rc+l\left(c^n\right)\subseteq L=l\left(e\right)$, so we have

$$Rce=0,$$

(40)

and thus, $ce=0$. Noting that $R$ is abelian, so we obtain

$$ec=ce=0.$$

(41)

Hence,

$$e\in l\left(c\right)\subseteq l\left(c^n\right)\subseteq L=l\left(e\right)$$

(42)

Thus, we obtain

$$e=e^2=0,$$

(43)

which contradicts the hypothesis. Therefore, $L$ is an essential ideal, and $R/L$ is a simple singular left $R$-module. Given that all maximal essential left ideals of $R$ are weakly right ideals, and $c^n\in L$, we derive $b{c}^n\in L$ for all $b\in R$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(44)

It is clear that $f$ qualifies as a well-defined left $R$-module homomorphism. By assumption, $R/L$ is an n-P-injective module, so there is $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L,$$

(45)

and it is obvious that $1-c^{n}b\in L$. By the fact that $L$ is a weakly right ideal and a left ideal, there is a positive integer $m$ satisfying ${\left(c^{n}b\right)}^m\in L$.

Noting that

$$\begin{gathered} 1-\left(c^{n}b\right)\in L, \\ {c}^{n}b(1-c^{n}b)=c^{n}b-(c^{n}b{)}^2\in L, \\ \left(c^{n}b{)}^2\right(1-c^{n}b)=(c^{n}b{)}^2-(c^{n}b{)}^3\in L, \\ \begin{array}{c}\dots \dots \dots \dots \end{array} \\ \left(c^{n}b{)}^{m-1}\right(1-c^{n}b)=(c^{n}b{)}^{m-1}-(c^{n}b{)}^m\in L. \end{gathered}$$

(46)

We can obtain $1-(a^{n}b{)}^m\in L,1\in L$. This is in conflict with the maximality of the ideal $L$. Thus, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Therefore, $R$ is a strong $\pi$-regular ring. □

Theorem 6.

The following are equivalent:

(1) 

$R$ is a strongly $\pi$-regular ring.

(2) 

$R$ is an abelian left n-P-V’-ring where all its maximal essential left ideals are generalized weak ideals.

(3) 

$R$ is an abelian right n-P-V’-ring where all its maximal essential right ideals are generalized weak ideals.

Proof. 

(1) ⇒ (2) and (1) ⇒ (3) are trivial. Since (3) ⇒ (1) is similar to (2) ⇒ (1), we only need (2) ⇒ (1) in the following.

For (2) ⇒ (1): Suppose there is an element c $\in R$ satisfying $Rc+l\left(c^n\right)\ne R$. Then the ring $R$ contains a maximal left ideal $L$ such that

$$Rc+l\left(c^n\right)\subseteq L\subset R.$$

(47)

In the following, we prove that $L$ is essential. Otherwise, $L$ is a direct sum term of the ring $R$, i.e., $L=l\left(e\right)$, where $e\ne 0, \mathrm{and} e^2=e\in R.$ Then $Rc+l\left(c^n\right)\subseteq L=l\left(e\right)$, so we have

$$Rce=0.$$

(48)

Thus, $ce=0$. Noting that $R$ is abelian, so we obtain

$$ec=ce=0.$$

(49)

Hence,

$$e\in l\left(c\right)\subseteq l\left(c^n\right)\subseteq L=l\left(e\right).$$

(50)

Thus, we obtain

$$e=e^2=0,$$

(51)

which contradicts the hypothesis. Therefore $L$ is an essential ideal, and $R/L$ is a simple singular left $R$-module. Given that all maximal essential left ideals of $R$ are generalized weak ideals, and $c^n\in L$, we have $b{c}^n\in L$ for all $b\in R$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(52)

It is clear that $f$ qualifies as a well-defined left $R$-module homomorphism. By assumption, $R/L$ is an n-P-injective module, so there is $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L.$$

(53)

It is obvious that

$$a^{n}b\in L,b\left(1-a^{n}b\right)=b-b{a}^{n}b\in L.$$

(54)

By the fact that $L$ is a generalized weak ideal and $b{c}^n\in L$, there is a positive integer $m$ satisfying ${\left(c^{n}b\right)}^{m}b\in L$. Hence,

$$(b{c}^n{)}^{m-1}b=(b{c}^n{)}^{m-1}(b-b{c}^{n}b)+(b{c}^n{)}^{m}b\in L,$$

(55)

Continuing this process, we can obtain $b{c}^{n}b\in L$. Since $b=\left(b-b{c}^{n}b\right)+b{c}^{n}b\in L$, this is in conflict with the maximality of the ideal $L$. Thus, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Therefore, $R$ is a strong $\pi$-regular ring. □

Subsequently, we specify three conditions that are sufficient to confer left weakly $\pi$-regularity upon idempotent reflexive n-P-V’-rings.

Theorem 7.

$R$ is reduced left weakly $\pi$-regular if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are quasi-ideals.

Proof. 

Suppose there is an element c $\in R$ satisfying $R{c}^{n}R+l\left(c^n\right)\ne R$. Then the ring $R$ contains a left ideal $L$ such that $R{c}^{n}R+l\left(c^n\right)\subseteq L\subset R$, and it is not hard to see that $L$ is an essential ideal. Indeed, if $L$ is not an essential ideal, $L$ is a direct sum term of the ring $R$. Set $L=l\left(e\right)$, where $e\ne 0,and{e}^2=e\in R$. We have $R{c}^{n}Re=0$, indicating $c^{n}Re=0$. Given that $R$ is an idempotent reflexive ring, we derive $eR{c}^n=0$ implying $e{c}^n=0$. Hence,

$$e\in l\left(c^n\right)\subseteq L=l\left(e\right)$$

(56)

Thus, we have $e=e^2=0$. This is in conflict with the assumption. Hence, $L$ is an essential ideal. By assumption, since $R/L$ is a left n-P-injective module, and all maximal essential left ideals of the ring $R$ are quasi-ideals, we have $c^n\in L$, and ${bc}^n\in L$ for all $b\in R$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(57)

According to Lemma 1, $R$ is a reduced ring, so $f$ qualifies as a well-defined left $R$-module homomorphism. Based on the assumption, $R/L$ is a left n-P-injective module, so there is $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L.$$

(58)

Hence, we have $1-c^{n}b\in L$ where $c^n\in l\left(c^n\right)\subseteq L$. Since $b{c}^n\in L$, and $L$ is a quasi-ideal and a left ideal, we derive

$$\left(1+b\right)c^n\left(1+b\right)\in L,b{c}^{n}b\in L,b{c}^n\in L.$$

(59)

Hence,

$$c^{n}b=\left(1+b\right)c^n\left(1+b\right)-c^n-b{c}^n-b{c}^{n}b\in L,$$

(60)

since $1=\left(1-c^{n}b\right)+c^{n}b\in L$, and this is in conflict with the maximality of the ideal $L$. Therefore, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Therefore, $R$ is a reduced left weakly π-regular ring. □

Theorem 8.

$R$ is reduced left weakly $\pi$-regular if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are weakly right ideals.

Proof. 

Suppose there is an element c $\in R$ such that $R{c}^{n}R+l\left(c^n\right)\ne R$. Then the ring $R$ contains a left ideal $L$ such that $R{c}^{n}R+l\left(c^n\right)\subseteq L\subset R$, and it is not hard to see that $L$ is an essential ideal. Indeed, if $L$ is not an essential ideal, $L$ is a direct sum term of the ring $R$. Set $L=l\left(e\right)$, where $e\ne 0,$ and $e^2=e\in R$. We have $R{c}^{n}Re=0$, indicating $c^{n}Re=0$. Given that $R$ is an idempotent reflexive ring, we derive $eR{c}^n=0$ implying $e{c}^n=0$. Hence,

$$e\in l\left(c^n\right)\subseteq L=l\left(e\right),$$

(61)

and thus, we have $e=e^2=0$. This is in conflict with the assumption. Hence, $L$ is an essential ideal. Based on the assumption, since $R/L$ is a left n-P-injective module, and all maximal essential left ideals of the ring $R$ are weakly right ideals, we have $c^n\in L$, and ${bc}^n\in L$ for all $b\in R$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(62)

According to Lemma 2, $R$ is a reduced ring, so $f$ qualifies as a well-defined left $R$-module homomorphism. Based on the assumption, $R/L$ is a left n-P-injective module, so there is $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L,$$

(63)

It’s obvious that $1-c^{n}b\in L$. Since $L$ is a weakly right ideal and a left ideal, there is a positive integer $m$ satisfying $(c^{n}b{)}^m\in L$. Therefore, we can derive

$$\begin{gathered} 1-\left(c^{n}b\right)\in L, \\ {c}^{n}b(1-c^{n}b)=c^{n}b-(c^{n}b{)}^2\in L, \\ \left(c^{n}b{)}^2\right(1-c^{n}b)=(c^{n}b{)}^2-(c^{n}b{)}^3\in L, \\ \begin{array}{c}\dots \dots \dots \dots \end{array} \\ \left(c^{n}b{)}^{m-1}\right(1-c^{n}b)=(c^{n}b{)}^{m-1}-(c^{n}b{)}^m\in L. \end{gathered}$$

(64)

Thus, we obtain $1-(c^{n}b{)}^m\in L$, $1\in L$. This is in conflict with the maximality of the ideal $L$. Therefore, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Therefore, $R$ is a reduced left weakly $\pi$-regular ring. □

Theorem 9.

$R$ is reduced left weakly $\pi$-regular if $R$ is an idempotent reflexive semi-abelian left n-P-V’-ring where all its maximal essential left ideals are generalized weak ideals.

Proof. 

Suppose there is an element $c\in R$ satisfying $R{c}^{n}R+l\left(c^n\right)\ne R$. Then the ring $R$ contains a left ideal $L$ satisfying $R{c}^{n}R+l\left(c^n\right)\subseteq L\subset R$, and it is not hard to see that $L$ is an essential ideal. Indeed, if $L$ is not an essential ideal, $L$ is a direct sum term of the ring $R$. Set $L=l\left(e\right)$, where $e\ne 0,$ and $e^2=e\in R$. We have $R{c}^{n}Re= 0$, indicating $c^{n}Re= 0$. Given that $R$ is an idempotent reflexive ring, we derive $eR{c}^n=0$ implying $e{c}^n=0$. Hence,

$$e\in l\left(c^n\right)\subseteq L=l\left(e\right),$$

(65)

and thus, we have $e=e^2=0$. This is in conflict with the assumption. Hence, $L$ is an essential ideal. Based on the assumption, since $R/L$ is a left n-P-injective module, and all maximal essential left ideals of the ring $R$ are generalized weak ideals, we have $c^n\in L$, and ${bc}^n\in L$ for all $b\in R$. We define

$$f:R{c}^n\to R/L,f\left(r{c}^n\right)=r+L,r\in R.$$

(66)

According to Lemma 3, $R$ is a reduced ring, so $f$ qualifies as a well-defined left $R$-module homomorphism. Based on the assumption, $R/L$ is a left n-P-injective module, so there is $b\in R$ satisfying

$$1+L=f\left(c^n\right)=c^{n}b+L.$$

(67)

It is obvious that

$$1-c^{n}b\in L,b\left(1-c^{n}b\right)=b-b{c}^{n}b\in L.$$

(68)

Since $b{c}^n\in L$, and $L$ is a generalized weak ideal, there is a positive integer $m$ satisfying $(b{c}^n{)}^{m}b\in L$. Thus, we obtain

$$(b{c}^n{)}^{m-1}b=(b{c}^n{)}^{m-1}(b-b{c}^{n}b)+(b{c}^n{)}^{m}b\in L$$

(69)

Continuing this process, we can obtain $b{c}^{n}b\in L$. Since $b=(b-b{c}^{n}b)+b{c}^{n}b\in L$, this is in conflict with the maximality of the ideal $L$. Hence, we derive $Rc+l\left(c^n\right)=R$ for all $c\in R$. Therefore, $R$ is a reduced left weakly $\pi$-regular ring. □

4. Conclusions and Prospects

This paper provides a comprehensive study of the regularity of n-P-V-rings and n-P-V’-rings. Utilizing the notions of quasi-ideals, weakly left/right ideals, and generalized weak ideals, the focus was primarily on investigating the strong $\pi$-regularity and weak $\pi$-regularity of n-P-V-rings and n-P-V’-rings.

Quasi-ideals, weakly left (or right) ideals, and generalized weak ideals are all extensions of the notion of ideals. According to the regularity theorems derived in this paper, it appears that these three entities can be interchanged under certain conditions. The deep intrinsic connections among these three types of ideals remain to be explored in future research.