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

Dariush Heidari∗; Bijan Davvaz

doi:10.5666/KMJ.2020.60.1.71

Article

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

Published online March 31, 2020 https://doi.org/10.5666/KMJ.2020.60.1.71

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

Abstract

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Abstract
1. Introduction
2. Main Result
References

A ring R is quasi-reversible if 0 ≠ ab ∈ I(R) for a, b ∈ R implies ba ∈ I(R), where I(R) is the set of all idempotents in R. In this short paper, we prove that the ring of 2×2 matrices over an arbitrary field is quasi-reversible, which is an answer to the question given by Da Woon Jung et al. in [Bull. Korean Math. Soc., 56(4) (2019) 993–1006].

Keywords: quasi-reversible ring, matrix ring.

1. Introduction

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Abstract
1. Introduction
2. Main Result
References

Let R be a ring. Use I(R) to denote the set of all idempotents in R and I(R)′ = I(R){0}. Let Mat_n(R) be n×n matrix ring over R. Following Da Woon Jung et. al. [1] a ring R is quasi-reversible provided that if ab ∈ I(R)′ for a, b ∈ R, then ba ∈ I(R).

Theorem 1.1

([1, Theorem 1.8]) Mat₂(ℤ₂) is quasi-reversible.

2. Main Result

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Abstract
1. Introduction
2. Main Result
References

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

Question 2.1

Let K be a field. Is Mat₂(K) quasi-reversible?

Theorem 2.2

Let K be a filed. Then Mat₂(K) is quasi-reversible.

Proof

Let K be a field and A,B ∈ Mat₂(K) such that AB ∈ I(Mat₂(K))′. If A is invertible, then

BA=A-1(AB)A=A-1(ABAB)A=BABA.

Hence BA ∈ I(Mat₂(K)). Similarly, if B is invertible, then BA is idempotent. It remains to consider the case that A and B are non-invertible. To this end we prove the following claims:

Claim 1

If M = (m_ij) is a singular 2 × 2 matrix, then M² = tr(M)M where, tr(M) = m₁₁ + m₂₂.

Proof of Claim 1

Since |M| = 0, it follows that m₁₁m₂₂ = m₁₂m₂₁. Consequently, we have

M2=(m112+m12m21m11m12+m12m22m11m21+m21m12m12m21+m222)=(m112+m11m22m12(m11+m22)m21(m11+m22)m11m22+m222)=(m11+m22)M=tr(M)M.

Claim 2

If M = (m_ij) is a non-zero singular and idempotent 2 × 2 matrix, then tr(M) = 1.

Proof of Claim 2

By Claim 1, we have 0 ≠ M = M² = tr(M)M thus (tr(M) – 1)M = 0 and hence tr(M) = 1.

Now, Let A and B be two non-invertible matrices such that AB ∈ I(Mat₂(K))′. Then, Claim 2 concludes tr(AB) = 1. Thus, by Claim 1 we deduce that

(BA)2=tr(BA)BA=tr(AB)BA=BA.

Therefore, BA is idempotent and the proof is completed.

References

Go to

Abstract
1. Introduction
2. Main Result
References

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.