### Article

Kyungpook Mathematical Journal 2021; 61(1): 11-21

**Published online** March 31, 2021

Copyright © Kyungpook Mathematical Journal.

### Structures Related to Right Duo Factor Rings

Hongying Chen, Yang Lee, Zhelin Piao∗

Department of Mathematics, Pusan National University, Pusan 46241, Korea

e-mail : 1058695917@qq.com

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, Yanbian University, Yanji 133002, China

e-mail : zlpiao@ybu.edu.cn

**Received**: December 9, 2019; **Revised**: June 30, 2020; **Accepted**: June 4, 2020

### Abstract

We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called *right FD*. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring *R* that *R* is a subdirect product of subdirectly irreducible right FD rings; and that *J(R)*) is the prime (Jacobson) radical of *R*. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring *R* is right FD and *eRe* is also right FD, examining that the class of right FD rings is not closed under subrings.

**Keywords**: right FD ring, right duo ring, division ring, commutative ring, simple ring, non-prime right FD ring, matrix ring, polynomial ring, subring, idempotent

### 1. Introduction

Throughout this note every ring is an associative ring with identity unless otherwise stated. Let _{*}(R)^{*}(R)_{n}

This article is motivated by the results in [9]. In Section 2 we study the structure of right FD rings, focusing on the relation among right FD rings, commutative rings and simple rings. We investigate that in several kinds of ring extensions that play important roles in ring theory. In Section 3 we examine the right FD property of polynomial rings, subrings and direct products for given right FD rings.

A ring is called ^{2}=0

### 2. When Factor Rings are Right Duo

In this section we are concerned with the class of rings whose factor rings modulo nonzero proper ideals are right duo. A ring

### Lemma 2.1.

(1) Every simple (or right primitive) right duo ring is a division ring.

(2) Every prime right (left) duo ring is a domain.

(3) The class of right (left) duo rings is closed under factor rings and direct products.

(4) If

R is a division ring thenD is a duo ring; but_{2}(R)D is neither right no left duo for all_{n}(A)n ≥ 3 over any ringA .

(5) Let

A be any ring andn ≥ 3 . ThenT is neither right nor left FD._{n}(A)

(6) Let

A be any ring andn ≥ 4 . ThenD is neither right nor left FD._{n}(A)

(7) The class of right (left) FD rings is closed under factor rings.

(2) Let

(3) is obvious.

(4) Take

(5) Note that

(6) Note that

(7) Let

The proofs for the left cases of (1)-(7) are similar.

Right duo rings are right FD by Lemma 2.1(3); but the converse is not true in general by the following. Note that _{n}(A)_{n}(A)_{n}(R)_{n}(R)

### Theorem 2.2.

(1)

R is simple if and only ifMat is right FD if and only if_{n}(R)Mat is simple._{n}(R)(2) The following conditions are equivalent:

(i)

R is a division ring;(ii)

T is a right (left) FD ring;_{2}(R)(iii)

D is a right (left) FD ring._{3}(R)

(3)] Let

R be simple. ThenR is a division ring if and only ifD is right (left) FD._{2}(R)

_{n}(R)_{n}(R/I)_{n}(R)

(2) We apply the proof of [9, Theorem 1.10(3)]. (i) _{2}(F)

(ii) _{2}(R)_{2}(R/M)_{2}(R)_{2}(R)_{2}(R)/I_{2}(R)

(i) _{3}(R)

respectively. Note that _{2}(R)

of

of

(iii) _{3}(R)_{3}(R)_{3}(R)_{3}(R)/I_{3}(R)

(3) It suffices to show the sufficiency by Lemma 2.1(4). Let _{2}(R)_{12}

Following [9], a ring _{2}(R)

Following Birkhoff [1], a ring

### Lemma 2.3.

Let

(1)

R is a subdirect product of subdirectly irreducible right (resp., left) FD rings.

(2)

R/N is a subdirect product of right (resp., left) duo domains, and_{*}(R)R/J(R) is a subdirect product of division rings.

(3) If

R is semiprime thenR is a subdirect product of right (resp., left) duo domains (hence reduced).

_{a}_{a}_{a}

(2) Let _{i}_{i}_{i}_{i}_{*}(R)

(3) is an immediate consequence of (2). The proofs of (1), (2) and (3) for the left case are similar.

There exist non-prime FD rings which are not subdirectly irreducible. In fact, each of

Based on Lemma 1.3, one may ask whether a ring

### Example 2.4.

We refer to the construction and argument in [7, Example 1.2] and [8, Theorem 2.2(2)]. Let _{n}_{n+1}

This implies that

Let

Following Neumann [12], a ring

### Proposition 2.5.

Let

(1)

R is right FD;

(2)

R is reduced;

(3)

R is right duo;

(4)

R is left duo;

(5)

R is left FD;

(6) Every right primitive factor ring of

R is a division ring;

(7)

R is a subdirect product of division ring;

(8)

R is a subdirect product of domains.

_{*}(R)=0

(1) ⇒ (6) is obtained from Lemma 2.1(1) because

The condition "non-prime" is not superfluous in Proposition 2.5 as can be seen by the regular ring _{n}(A)_{n}(A)

Following [14], a ring

### Proposition 2.6.

(1) Every non-prime right FD ring is right quasi-duo.

(2) If

R is a non-prime right FD ring thenR/J(R) is a reduced right quasi-duo ring.

(2) is obtained from (1) and Lemma 2.3(2).

The following elaborates upon Proposition 2.6.

### Remark 2.7.

(1) Simple (hence FD) rings need not be quasi-duo by the existence of simple domains which are not division rings (e.g., the first Weyl algebra over a field of characteristic zero), which is compared with Proposition 2.6(1). Indeed this domain is neither right nor left quasi-duo.

(2) There exist non-prime noncommutative FD rings as can be seen by

T and_{2}(R)D over a division ring_{3}(R)R (see Theorem 2.2(2)). This provides examples to Proposition 2.6.

(3) Based on Proposition 2.6(1), one may ask whether a non-prime right quasi-duo ring is right FD. But the answer is negative. Let

A be a right quasi-duo ring andR=T for_{n}(A)n ≥ 3 . ThenR is right quasi-duo by [14, Proposition 2.1]. LetI=AE . Then_{1n}R/I is non-Abelian (hence not right duo), and soR is not right FD.

Next we will show that the FD property is not left-right symmetric.

### Example 2.8.

Consider a skewed trivial extension in [13, Definition 1.3] as follows. Let ^{2})^{2}

Now let _{1}=K

### 3. Subrings, Polynomial Rings and Direct Products

In this section we study the right FD property of polynomial rings, subrings and direct products of given right FD rings. We consider first the polynomial ring case.

### Theorem 3.1.

The following conditions are equivalent for a given ring

(1)

R[x] is right (left) FD;

(2)

R is commutative;

(3)

R[x] is commutative.

from the nonzero proper ideal

We can show, by help of Theorem 3.1, that the right FD property does not pass to polynomial rings.

We can write the following by help of Theorem 3.1 and Lemma 2.1(3): For a ring

We next argue about subrings of right (left) FD rings.

### Example 3.2.

(1) Let

R be the first Weyl algebra over a field of characteristic zero. ConsiderR[x] . SinceR[x] is right Noetherian domain, there exists the quotient division ring,Q say.Q is clearly FD. ButR[x] is noncommutative, henceR[x] is neither right nor left FD by Theorem 3.1.

(2) We extend (1). Let

D be any right Noetherian domain that is not a division ring. LetQ be the quotient division ring. ThenT is FD by Theorem 2.2(2). But the subring_{2}(Q)T is neither right nor left FD by Theorem 2.2(2) because_{2}(D)D is not a division ring.

In the following we find a kind of subring which inherits the right FD property.

### Theorem 3.3.

(1) Let

R be a ring and0 ≠ e . If^{2}=e ∈ RR is right (resp., left) duo theneRe is right (resp., left) duo.

(2) Let

R be a ring and0 ≠ e . If^{2}=e ∈ RR is right (resp., left) FD theneRe is right (resp., left) FD.

(2) We apply the proof of [9, Theorem 1.12]. Suppose that

Suppose that

Write

Since

Let

Next let

Recall that right duo rings are right FD. In contrast to Lemma 2.1(3), one may ask whether the direct product of right FD rings is also right FD. But the answer is negative as follows. Let

In the following we see an equivalent condition for direct products of right FD rings to be right FD.

### Theorem 3.4.

Let _{i}

(1)

R is right FD;

(2)

R is right duo for all_{i}$i\in I$ ;

(3)

R is right duo.

_{j}_{j}_{j}_{j}

(3) ⇒ (1) is obvious, and (2) ⇒ (3) is shown by Lemma 2.1(3).

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