Article
Kyungpook Mathematical Journal 2021; 61(3): 461-472
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
On Axis-commutativity of Rings
Tai Keun Kwak, Yang Lee, Young Joo Seo*
Department of Mathematics, Daejin University, Pocheon 11159, Korea
e-mail : tkkwak@daejin.ac.kr
Department of Mathematics, Yanbian University, Yanji 133002, China and Institute of Basic Science, Daejin University, Pocheon 11159, Korea
e-mail : ylee@pusan.ac.kr
Department of Mathematics, Daejin University, Pocheon 11159, Korea
e-mail : jooggang@daejin.ac.kr
Received: August 23, 2020; Revised: April 29, 2020; Accepted: May 18, 2021
Abstract
We study a new ring property called axis-commutativity. Axis-commutative rings are seated between commutative rings and duo rings and are a generalization of division rings. We investigate the basic structure and several extensions of axis-commutative rings.
Keywords: division ring, axis-commutative ring, commutative ring, duo ring, matrix ring.
1. Introduction
Throughout this note every ring is an associative ring with identity unless otherwise stated. Let
It is well-known that
A ring is usually called
Recall that for a ring
with the usual addition and the following multiplication:
This is isomorphic to the ring of all matrices
The study of the trivial extension of generalized reduced rings plays a significant role in noncommutative ring theory to understand the ring structure. For example, the trivial extension of a reduced ring is not reduced but contained in some class of generalized reduced rings. In addition, a ring
Theorem 1.1. Let
and compute the relation between
(Case 1) Suppose
(Case 2) Suppose
(Case 3) Suppose
(Case 4) Suppose
Summarizing, we conclude that
Example 1.2. (1) The condition '
For
(2) The converse of Theorem 1.1 does not hold, in general. For example, consider a commutative ring
Based on the above, we define a new ring property as follows.
Definition 1.3. A ring
Then we obtain the next results.
Proposition 1.4. (1) If the trivial extension
(2)
(3) Both
Then
Similarly, it can be obtained
Consequently,
(2) Let
(3) Let
The computation for
Let
Proposition 1.5. Let
(1) If
(2) If the ring
(1) Suppose that
If
If
Now we show that
We note that
With the help of
Inductively we assume that
Consequently,
Since
(2) Suppose that
Recall that
The next example shows that the converse of Proposition 1.4(1) (also Proposition 1.5(2)) need not hold.
Example 1.6. Let
Then
and
This implies that
2. Property of Axis-commutative Rings
Following Feller [4], a ring (possibly without identity) is called
An axis-commutative ring is a generalization of division rings as noted in Section 1. In this section, we show that the class of axis-commutative rings is seated between commutative rings and duo rings, and investigate the basic structure and several extensions of axis-commutative rings.
In the next lemma we observe basic properties of an axis-commutative ring which do important roles throughout this article.
Lemma 2.1. (1) A ring
(2) Axis-commutative rings are duo (hence, Abelian).
(3) If
(4) For a ring
(5) Let
(6) The class of axis-commutative rings is closed under homomorphic images.
(2) Let
(3) Suppose that
(4) The proof is clear.
(5) Suppose that
Conversely, suppose that every
(6) Suppose that
Observe that
As corollaries of Lemma 2.1(2, 5), we have the following.
Corollary 2.2. (1) A ring
(2) Let
The converse is clear by Lemma 2.1(2) and definition.
(2) It comes from the facts
The following example shows that the converse of Lemma 2.1(2) does not hold as well as the condition '
Example 2.3. We follow the construction and argument in [15, Example 2]. Let
Set
We will show that
because
when
These entail
The following example shows that the converse of Lemma 2.1(6) does not hold. That is there exists a ring
Example 2.4. Consider
A ring
Proposition 2.5. A ring
Conversely assume that
As a generalization of a reduced ring, Cohn [2] called a ring
Remark 2.6. Let
(1) Then
(2) If
(3) If
Following Goodearl [5], a ring
Due to [13], a right ideal
Proposition 2.7 (1) Every axis-commutative ring is reversible, and hence it is reflexive.
(2) Let
If
(2) Suppose that
Remark 2.8. (1) Notice that there exists a domain which is not axis-commutative. Recall the domain
(2) Related to Proposition 2.7(1), note that (i) there exists an axis-commutative ring which is not reduced (and hence not a domain) by help of Theorem 1.1, i.e., the condition '
The next example illuminates that the converse of Proposition 2.7(1) does not hold.
Example 2.9. (1) For a reflexive ring
(2) Consider the ring
be the free algebra with noncommuting indeterminates
over
where the constant terms of
For an algebra
Proposition 2.10. Let
Thus we have
Conversely, assume that
Recall that when
Proposition 2.11. For a field
(1)
(2)
(3)
(4)
(5)
(6)
(7) The equation
Theorem 2.12. For a ring
(1)
(2)
(3)
(4)
(5)
Observe that if the polynomial ring
Acknowledgements.
The authors thank the referee deeply for very careful reading of the manuscript and valuable suggestions in depth that improved the paper by much.
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