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 : firstname.lastname@example.org
Department of Mathematics, Yanbian University, Yanji 133002, China and Institute of Basic Science, Daejin University, Pocheon 11159, Korea
e-mail : email@example.com
Department of Mathematics, Yanbian University, Yanji 133002, China
e-mail : firstname.lastname@example.org
Received: December 9, 2019; Revised: June 30, 2020; Accepted: June 4, 2020
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
Keywords: right FD ring, right duo ring, division ring, commutative ring, simple ring, non-prime right FD ring, matrix ring, polynomial ring, subring, idempotent
Throughout this note every ring is an associative ring with identity unless otherwise stated. Let
This article is motivated by the results in . 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. 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
(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.
Ris a division ring then D2(R)is a duo ring; but Dn(A)is neither right no left duo for all n ≥ 3over any ring A.
Abe any ring and n ≥ 3. Then Tn(A)is neither right nor left FD.
Abe any ring and n ≥ 4. Then Dn(A)is neither right nor left FD.
(7) The class of right (left) FD rings is closed under factor rings.
(3) is obvious.
(5) Note that
(6) Note that
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
Ris simple if and only if Matn(R)is right FD if and only if Matn(R)is simple.
(2) The following conditions are equivalent:
Ris a division ring;
T2(R)is a right (left) FD ring;
D3(R)is a right (left) FD ring.
Rbe simple. Then Ris a division ring if and only if D2(R)is right (left) FD.
(2) We apply the proof of [9, Theorem 1.10(3)]. (i)
respectively. Note that
(3) It suffices to show the sufficiency by Lemma 2.1(4). Let
Following , a ring
Following Birkhoff , a ring
Ris a subdirect product of subdirectly irreducible right (resp., left) FD rings.
R/N*(R)is a subdirect product of right (resp., left) duo domains, and R/J(R)is a subdirect product of division rings.
Ris semiprime then Ris a subdirect product of right (resp., left) duo domains (hence reduced).
(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
We refer to the construction and argument in [7, Example 1.2] and [8, Theorem 2.2(2)]. Let
This implies that
Following Neumann , a ring
Ris right FD;
Ris right duo;
Ris left duo;
Ris left FD;
(6) Every right primitive factor ring of
Ris a division ring;
Ris a subdirect product of division ring;
Ris a subdirect product of domains.
(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
Following , a ring
(1) Every non-prime right FD ring is right quasi-duo.
Ris a non-prime right FD ring then R/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.
(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
T2(R)and D3(R)over a division ring 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
Abe a right quasi-duo ring and R=Tn(A)for n ≥ 3. Then Ris right quasi-duo by [14, Proposition 2.1]. Let I=AE1n. Then R/Iis non-Abelian (hence not right duo), and so Ris not right FD.
Next we will show that the FD property is not left-right symmetric.
Consider a skewed trivial extension in [13, Definition 1.3] as follows. Let
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.
The following conditions are equivalent for a given ring
R[x]is right (left) FD;
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.
Rbe the first Weyl algebra over a field of characteristic zero. Consider R[x]. Since R[x]is right Noetherian domain, there exists the quotient division ring, Qsay. Qis clearly FD. But R[x]is noncommutative, hence R[x]is neither right nor left FD by Theorem 3.1.
(2) We extend (1). Let
Dbe any right Noetherian domain that is not a division ring. Let Qbe the quotient division ring. Then T2(Q)is FD by Theorem 2.2(2). But the subring T2(D)is neither right nor left FD by Theorem 2.2(2) because Dis not a division ring.
In the following we find a kind of subring which inherits the right FD property.
Rbe a ring and 0 ≠ e2=e ∈ R. If Ris right (resp., left) duo then eReis right (resp., left) duo.
Rbe a ring and 0 ≠ e2=e ∈ R. If Ris right (resp., left) FD then eReis right (resp., left) FD.
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.
Ris right FD;
Riis right duo for all ;
Ris right duo.
(3) ⇒ (1) is obvious, and (2) ⇒ (3) is shown by Lemma 2.1(3).
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