### Article

Kyungpook Mathematical Journal 2024; 64(1): 15-30

**Published online** March 31, 2024

Copyright © Kyungpook Mathematical Journal.

### Generalized Inverses and Solutions to Equations in Rings with Involution

Yue Sui^{∗} and Junchao Wei

Department of Mathematics, Yangzhou University,Yangzhou, 225002, P. R. China

e-mail : suiyue052@126.com and jcweiyz@126.com

**Received**: May 4, 2021; **Revised**: June 23, 2022; **Accepted**: July 25, 2022

### Abstract

In this paper, we focus on partial isometry elements and strongly EP elements or a ring. We construct characterizing equations such that an element which is both group invertible and MP-invertible, is a partial isometry element, or is strongly EP, exactly when these equations have a solution in a given set. In particular, an element *a* ∈ *R*^{#} ∩ *R*^{†} is a partial isometry element if and only if the equation *x* = *x*(*a*^{†})**a*^{†} has at least one solution in {*a*, *a*^{#}, *a*^{†}, *a**, (*a*^{#})*, (*a*^{†})*}. An element *a* ∈ *R*^{#}∩*R*^{†} is a strongly EP element if and only if the equation (*a*^{†})**xa*^{†} = *xa*^{†}*a* has at least one solution in {*a*, *a*^{#}, *a*^{†}, *a**, (*a*^{#})*, (*a*^{†})*}. These characterizations extend many well-known results.

**Keywords**: EP element, Normal EP element, Strongly EP element, Partial isometry

### 1. Introduction

Throughout this paper, *R* denotes an associative ring with 1. We write *E*(*R*) and *J*(*R*) to denote the set of all idempotents and the Jacobson radical of *R*, respectively.

An element

The element *a*, which is uniquely determined by the above equations [1, 9].

An involution *R* is an anti-isomorphism of degree 2, that is,

An element *a* in *R* is called normal if

An element *R* is called the Moore-Penrose inverse (MP-inverse) of *a* [6, 10], if

If such *R* and the set of all MP-invertible elements of *R*, respectively.

An element *a* is said to be EP if *R*. Note that if *a* is normal and *R*.

An element *a* is called a partial isometry if *a* is called a strongly EP element if *R* by

In [7], D. Mosi*a* in a ring with involution to be a partial isometry are given. Recent researches on EP elements in rings with involution have produced some interesting results, see [4, 9, 12]. The necessary and sufficient conditions for the existence of a common solution and the general common solution of the equation *axb*=*c* (

Motivated by these articles above, this paper is intended to provide, by using certain equations admitting solutions in a definite set, further sufficient and necessary conditions for an element in a ring with involution to be an EP element, partial isometry, normal EP element, and strongly EP element. This is a new way to study generalized inverses in rings.

### 2. EP elements

**Lemma 2.1.[5, 7]** *Let *

*Proof.* Pre-multiplying the equality

**Lemma 2.2.** *Let *

*Proof.* ⇒ The equality obviously holds since

⇐ Post-multiplying *a*, one has

In order to prove the theorems given in this paper more clearly, we briefly review the following existing conclusions:

**Lemma 2.3.[11, Lemma 2.2]** *Let *

**Lemma 2.4.[11, Lemma 2.3]** *Let *

(1)

(2)

**Lemma 2.5.[12, Theorem 3.9]** *Let *

(1)

(2)

(3)

(4)

(5)

**Lemma 2.6.[14, Lemma 2.1]** *Let *

(1)

(2)

(3)

(4)

In [14, Theorem 2.4], the authors proved that an element

Recall that an element *a* is said to be EP if

**Theorem 2.7.** *Let *

*Proof.* ⇒ Obviously,

⇐ (1) If *x*=*a* is a solution, then

Therefore

(2) If

The fact that

(3) If

Since

(4) If

(5) If

(6) If

Lemma 2.6 now leads to

From Lemma 2.5,

Multipying the equation (2.1) on the right by *a*, we obtain:

**Theorem 2.8.** *Let *

*Proof.* ⇒Obviously,

⇐ (1) If *x*=*a* is a solution, then

Hence

(2) If *a*, we obtain that

By Lemma 2.6, we get

Therefore,

(3) If *a*, we have

by Lemma 2.6.

And then it follows from Lemma 2.5 that

(4) If

Hence

(5) If

which yields

(6) If

Therefore,

Further, we revised the equation (2.2) as follows:

**Theorem 2.9.** *Let *

*Proof.* ⇒

⇐ (1) If *x*=*a* is a solution, then

(2) If

(3) If

(4) If

By Lemma 2.5,

(5) If

which gives

(6) If

Pre-multiplying it by

Hence

Multiplying the last equality by

Post-multiplying it by

**Theorem 2.10.** *Let *

*Proof.* ⇒ Since

⇐ Assume that

**Theorem 2.11.** *Let *

*Proof.* ⇒ Since

⇐ Assume that

Pre-multiplying (2.4) by *a*, we obtain that

That is

On the other hand, post-multiply (2.4) by *a*, we have

Pre-multiplying (2.5) by *a* and then post-multiplying the last equation by

That is

Let *R*, which is called the right annihilator of *a*. Similarly, we can define

**Theorem 2.12.** *Let *

*Proof.* ⇒ Since

⇐ Assume that

### 3. Partial Isometry Elements

Recall that an element *R*, we have the following theorem.

**Theorem 3.1.** *Let *

(1)

(2)

*Proof.* The equality

On the contrary, assume that

Since

**Theorem 3.2.** *Let *

(1)

(2)

(3)

(4)

(5)

(6)

(7)

*Proof.* (1)⇒(2) From the assumption, we know that *a* and then post-multiplying the last equality by

(2)⇒(3) From (2), we obtain that *a* on the right and then we get

(3)⇒(4) Since

(4)⇒(5) Write

(5)⇒(6) From (5), we know that *a* and then multiplying the last equality by

Hence

(6)⇒(7) From (6), we obtain that

(7)⇒(1) Similar to (4)⇒(5), we obtain that

Therefore,

**Lemma 3.3.** *Let *

*Proof.* ⇒ Since

⇐ From the assumption, we know that *a* and then post-multiplying the last equality by *a*, we get

**Lemma 3.4.** *Let *

*Proof.* ⇒ We know that

⇐ From the assumption, we know that

**Lemma 3.5.** *Let *

*Proof.* ⇒ Obviously,

⇐ (1) If *x*=*a* is a solution, then *a*, we have

(2) If

(3) If

(4) If

(5) If *a*, we get

(6) If *a*, apply involution to the latest equation, and then we get

Similarly, we have the following theorem.

**Theorem 3.6.** *Let *

Using the symmetricity, we have the following corollary.

**Corollary 3.7.** *Let *

### 4. Normal EP Elements

**Lemma 4.1.** *Let *

(1) If

(2) If

*Proof.* (1) Pre-multiplying the equality

(2) Similarly, we can prove (2).

**Lemma 4.2.** *Let *

*Proof.* Since

**Lemma 4.3.** *Let *

*Proof.* Post-multiplying

**Lemma 4.4.** *Let *

*Proof.* Pre-multiplying

**Lemma 4.5.[14, Lemma 2.3]** *Let *

(1)

(2)

(4)

(6)

(3)

(5)

**Lemma 4.6.[14, Lemma 2.11]** *Let *

**Theorem 4.7.** *Let *

*Proof.* ⇒ By [11, Corollary 2.8], we know that *x*=*a* is a solution.

⇐ (1) If *x*=*a* is a solution, then *a*, we get *a* is normal. Hence

(2) If

(3) If

which shows

(4) If

(5) If

(6) If

Similar to the proof of (3),

**Theorem 4.8.** *Let *

*Proof.* ⇒ Since *x*=*a* is a solution.

⇐ (1) If *x*=*a* is a solution, then

Hence

Therefore,

(2) If

By Lemma 4.2 and the proof of (1),

Post-multiplying *a*, we have

(3) If

By Lemma 4.1,

(4) If

(5) If

By Lemma 4.3,

which implies

(6) If

### 5. Strongly EP elements

**Theorem 5.1.** *Let *

*Proof.* ⇒ Note that

⇐ (1) If *x*=*a* is a solution, then *a*, one gets

which leads to

(2) If

(3) If

Then,

(4) If

Applying the involution to the equality, one has

which gives

(5) If

By (4), we get

(6) If

**Theorem 5.2.** *Let *

*Proof.* ⇒ Obviously *x*=*a* is a solution since

⇐ (1) If *x*=*a* is a solution, then

By Lemma 4.5, *a*, we get

(2) If

(3) If

(4) If

(5) If

This implies

(6) If *a*, we get

**Theorem 5.3.** *Let *

*Proof.* ⇒ *x*=*a* is a solution since

⇐ (1) If *x*=*a* is a solution, then *a*, we obtain that

(2) If

This implies that

(3) If

Post-multiplying by

Hence

(4) If

Therefore *a*, we have

(5) If

By Lemma 4.5,

(6) If

**Theorem 5.4.**

*Proof.* ⇒

⇐ (1) If *x*=*a* is a solution, then

(2) If *a*, we get *a*, we obtain that

which shows

(3) If

(4) If

(5) If

(6) If

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