### Article

Kyungpook Mathematical Journal 2020; 60(1): 71-72

**Published online** March 31, 2020

Copyright © Kyungpook Mathematical Journal.

### Quasi-reversibility of the Ring of 2×2 Matrices over an Arbitrary Field

Dariush Heidari∗, Bijan Davvaz

Faculty of science, Mahallat Institute of Higher Education, Mahallat, Iran

e-mail : dheidari82@gmail.com

Department of Mathematics, Yazd University, Yazd, Iran

e-mail : davvaz@yazd.ac.ir

**Received**: September 14, 2019; **Accepted**: January 29, 2020

A ring

**Keywords**: quasi-reversible ring, matrix ring.

### 1. Introduction

Let _{n}

### Theorem 1.1

([1, Theorem 1.8]) _{2}(ℤ_{2})

### 2. Main Result

Now, we propose an answer to Question 1 stated in Da Woon Jung et. al. [1].

### Question 2.1

Let _{2}(

### Theorem 2.2

_{2}(

**Proof**

Let _{2}(_{2}(

Hence _{2}(

**Claim 1**

If _{ij}^{2} = _{11} + _{22}.

**Proof of Claim 1**

Since |_{11}_{22} = _{12}_{21}. Consequently, we have

### Claim 2

If _{ij}

**Proof of Claim 2**

By Claim 1, we have 0 ≠ ^{2} =

Now, Let _{2}(

Therefore,

- DW. Jung, CI. Lee, Y. Lee, S. Park, SJ. Ryu, HJ. Sung, and SJ. Yun.
On reversibility related to idempotents . Bull Korean Math Soc.,56 (4)(2019), 993-1006.