### 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 _{*}(R)^{*}(R)

It is well-known that _{n}(R)_{ij}

A ring is usually called ^{2=0}

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) _{n}(R)

(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 ^{2}=0

With the help of _{2}_{2}_{3}

Inductively we assume that

Consequently,

Since

(2) Suppose that

Recall that ^{n})^{n}

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 ^{2=0}^{3=0}

(5) Let

(6) The class of axis-commutative rings is closed under homomorphic images.

(2) Let

(3) Suppose that ^{2}=e∈ R

(4) The proof is clear.

(5) Suppose that

Conversely, suppose that every

(6) Suppose that

Observe that _{n}(R)_{n}(R)

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 _{1}=1

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 _{3}(F)

A ring

**Proposition 2.5.** A ring

_{i}R

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 ^{2 =0}

**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 _{1}_{1}

For an algebra _{i}∈ R_{i} ∈ S

**Proposition 2.10.** Let

Thus we have

Conversely, assume that

Recall that when

**Proposition 2.11.** For a field _{8}_{8}

(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.

### References

- H. E. Bell, Y. Li, Duo group rings, J. Pure Appl. Algebra, 209(2007), 833-838.
- P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(1999), 641-648.
- J. L. Dorroh, Concerning adjunctions to algebras, Bull. Amer. Math. Soc., 38(1932), 85-88.
- E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc., 89 (1958), 79-91.
- K. R. Goodearl, Von Neumann Regular Rings, Pitman, London (1979).
- C. Huh, H. K. Kim, Y. Lee, p.p. rings and generalized p.p. rings, J. Pure Appl. Algebra, 167(2002), 37-52.
- N. K. Kim, Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra, 185(2003), 207-223.
- T. K. Kwak, Y. Lee, Reflexive property of rings, Comm. Algebra, 40(2012), 1576-1594.
- J. Lambek, Lectures on Rings and Modules, Blaisdell Publishing Company, Waltham, 1966.
- G. Marks, Duo rings and Ore extensions, J. Algebra, 280(2004), 463-471.
- G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra, 174(2002), 311-318.
- C. P. Milies, S.K. Sehgal, An Introduction to Group Rings, Kluwer Academic Publishers, Dordrecht, 2002.
- G. Mason, Reflexive ideals, Comm. Algebra, 9(1981), 1709-1724.
- L. Motais de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents. Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), Bib. Nat., Paris (1982), 71-73.
- W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings, Comm. Algebra, 20(1992), 2777-2788.