Article
Kyungpook Mathematical Journal 2023; 63(2): 175186
Published online June 30, 2023 https://doi.org/10.5666/KMJ.2023.63.2.175
Copyright © Kyungpook Mathematical Journal.
Weakly Right IQNN Rings
Yang Lee, Sang Bok Nam, Zhelin Piao∗
Department of Mathematics, Yanbian University, Yanji 133002, P. R. China and Institute for Applied Mathematics and Optics, Hanbat National University, Daejeon 34158, Korea
email : ylee@pusan.ac.kr
Department of Computer Engineering, Kyungdong University, Geseong 24764, Korea
email : k1sbnam@kduniv.ac.kr
Department of Mathematics, Yanbian University, Yanji 133002, P. R. China
email : zlpiao@ybu.edu.cn
Received: June 11, 2022; Accepted: February 7, 2023
In this article we look at the property of a 2 by 2 full matrix ring over the ring of integers, of being
Keywords: weakly right IQNN ring, idempotent, nilpotent, 2 by 2 full matrix ring, 2 by 2 upper triangular matrix ring, right IQNN ring
Throughout this article every ring is an associative ring with identity unless otherwise stated. Let
Following Kwak et al. [5, Definition 1.2], a ring
The following facts have essential role in this article.
Define the sets
The following is from [5, Lemma 2.3(2, 3)].
Lemma 1.1. If
The following is from [1, Lemma 2.1(2)].
Lemma 1.2. Let
from which it follows that if
we have that if
In the following, we see a practical application of Lemma 1.2 that may provide useful information to the studies related to products of idempotents and nilpotents.
Remark 1.3. Let
(1) Let
or
That is,
We will find
Case 1. Suppose that
Let
Since
noting
and
noting
Thus
entailing
Case 2. The results in this case are obtained by the argument of [1, Lemma 2.1(3)].
(i) Suppose that
(ii) Suppose that
(iii) Suppose that
Thus there exist
(2) Let
or
that is,
We will find
Case 1. Suppose that
By applying the argument and using the notation of (1), we have
noting
and
noting
Thus
entailing
Case 2. The results in this case are obtained by the argument of [1, Lemma 2.1(3)].
(i) Suppose that
(ii) Suppose that
(iii) Suppose that
Thus there exist
(3) Let
We will find
Case 1. Suppose that
We have
noting
Based on Case 1, we can assume that
Case 2. Suppose that
We have
noting
Case 3. Suppose that
Let
Thus
(4) Let
Then we have the cases that
We will find
Case 1. Suppose that
We have
noting
Based on Case 1, we assume
Case 2. Suppose that
We have
noting
Case 3. Suppose that
Let
Thus
(5) Let
Let
Then we have the case that
Let
Then we have the case that
Let
Then we have the case that
Let
Then we have the case that
(6) Let
By (1) and (2), there exist
By (3) and (4), there exist
Similar arguments are available to the cases of
2. Weakly Right IQNN Rings
Motivated by the arguments of Remark 1.3., we consider the following new ring property as a generalization of right IQNN ring.
Definition 2.1. A ring
(i) There exist
(ii) There exist
(i) There exist
(ii) There exist
A ring is
Right IQNN rings are clearly weakly right IQNN, but not conversely as we see in the arguments below.
Theorem 2.2.
and
Then we have
Thus
In the following, we see another kind of weakly right IQNN rings but not right IQNN.
Example 2.3. Let
(1) We use the ring of the ring of [4, Example 2.3(2)]. Let
By applying the arguments of [4, Example 2.3(1)] and [5, Example 2.6], we have the following:
(i) every element
(ii)
(iii)
Since
where
(2) Let
(3) Let
(i) Every element
(ii)
(iii)
Since
where
(4) Let
The nonAbelian rings of Example 2.3 are all weakly IQNN. Next we provide a method by which one can construct nonAbelian rings that are neither weakly right nor weakly left IQNN.
Example 2.4. We use the ring of [3][Example 1.2(2)]. Let
where
Let
a contradiction. Next assume that
and
a contradiction. Thus
Next we consider two kinds of rings
Proposition 2.5. Let

(1) If
N(R)=N^{*}(R) thenT_{2}(R) is weakly right IQNN. 
(2) If
I(R)={0, 1} thenT_{2}(R) is weakly right IQNN.
and
(1) Assume
(2) Assume
Let
In the following argument we see a condition under which the weakly IQNN property is rightleft symmetric. Let
Proposition 2.6. Let
(i) There exist
(ii) There exist
Since
Conversely suppose that
(iii) There exist
(iv) There exist
Since
The second named author was supported by Kyungdong University Research Fund, 2022.
The third named author was supported by the Science and Technology Research Project of Education Department of Jilin Province, China(JJKH20210563KJ).
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