Article
Kyungpook Mathematical Journal 2020; 60(2): 307318
Published online June 30, 2020
Copyright © Kyungpook Mathematical Journal.
Rings which satisfy the Property of Inserting Regular Elements at Zero Products
Hong Kee Kim, Tai Keun Kwak*, Yang Lee, Yeonsook Seo
Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea
email : hkkim@gsnu.ac.kr
Department of Mathematics, Daejin University, Pocheon 11159, Korea
email : tkkwak@daejin.ac.kr
Department of Mathematics, Yanbian University, Yanji 133002, China and Institute of Basic Science, Daejin University, Pocheon 11159, Korea
email : ylee@pusan.ac.kr
Department of Mathematics, Pusan National University, Busan 46241, Korea
email : ysseo0305@pusan.ac.kr
Received: October 22, 2019; Revised: January 21, 2020; Accepted: February 10, 2020
Abstract
This article concerns the class of rings which satisfy the property of inserting regular elements at zero products, and rings with such property are called
Keywords: regularIFP ring, regular element, IFP ring, polynomial ring, generalized reduced ring
1. Introduction
Throughout this article every ring is an associative ring with identity. Let
2. RegularIFP Rings
In this section we study the properties of regularIFP rings as well as the relations between regularIFP rings and ring properties that play important roles in noncommutative ring theory. Due to Bell [4], a ring is called
Definition 2.1
A ring
RegularIFP rings are clearly unitIFP (hence Abelian), but not conversely by the following example.
Example 2.2
There exists a unitIFP ring that is not regularIFP. Let
Remark 2.3

(1) The following conditions are equivalent, which can be proved by applying the regularIFPness iteratively:

(i) A ring
R is regularIFP; 
(ii)
a _{1}C (R )a _{2}C (R )a _{3} ⋯a _{n} _{−1}C (R )a _{n} = 0 whenevera _{1}a _{2} ⋯a _{n} = 0 fora _{1},a _{2}, …,a _{n} ∈R .


(2)
D _{3}(R ) is (regular)IFP over a reduced ringR by [15, Proposition 2.1]. HoweverM _{n} (R ) andT _{n} (R ), over any ringR forn ≥ 2, cannot be regularIFP since they are not Abelian, noting that unitIFP (or regularIFP) rings are Abelian. 
(3) There exists an Abelian ring that is not regularIFP. Set
R =D _{n} (S ) forn ≥ 4 over an Abelian ringS . ThenR is Abelian by [10, Lemma 2]. LetA =E _{12},B =E _{34} ∈R . ThenAB = 0. ConsiderC =I _{n} +E _{23} ∈R . ThenC ∈C (R ) clearly. ButACB =E _{14} ≠ 0, so thatR is not regularIFP. 
(4) Let
R be a regularIFP ring such thatR =C (R ) ∪N (R ). Letab = 0 fora ,b ∈R . ThenaC (R )b = 0 sinceR is regularIFP. MoreoveraN (R )b = 0 by [15, Lemma 1.2(2)]. ThusR is IFP.
Based on Armendariz [3, Lemma 1], a ring
By Goodearl [7], a ring
Following [11], a ring is called
Proposition 2.4

(1)
Let R be a locally finite ring .
(i)
If R is an Armendariz ring, then it is regularIFP . 
(ii)
If R is a regularIFP ring, then R /J (R )is strongly regular with J (R ) =N (R ).


(2)
Let R be a right or left Artinian ring. If R is regularIFP, then R /J (R )is a strongly regular ring with J (R ) =N (R ).
(1)–(i) Let
(1)–(ii) Since regularIFP rings are Abelian, we obtain the result by [11, Proposition 2.5].
(2) It is wellknown that
The class of regularIFP rings is not closed under homomorphic images as can be seen by the ring
Proposition 2.5

(1)
Let R _{λ} (λ ∈ Λ) be Abelian rings. Then R _{λ} is regularIFP for each λ ∈ Λif and only if Π_{λ} _{∈Λ}R _{λ} is regularIFP if and only if the subring of Π_{λ} _{∈Λ}R _{λ} generated by ⊕_{λ} _{∈Λ}R _{λ} and 1_{Πλ∈Λ}_{R} _{λ}is regularIFP . 
(2)
Let R be an Abelian ring and e ^{2} =e ∈R. Then R is regularIFP if and only if both eR and (1 −e )R are regularIFP .
(1) Suppose that the subring of Π
(2) The proof is obtained from (1) since
For a given ring
Corollary 2.6
Suppose that
Conversely assume that
Recall that homomorphic images of regularIFP rings need not be regularIFP. Considering this fact, one may ask whether a ring
Example 2.7
There exists a nonregularIFP ring
3. Extensions of RegularIFP Rings
In this section we examine the regularIFP property of ring extensions that play roles in noncommutative ring theory.
Regarding Remark 2.3(3), we have the following.
Proposition 3.1

(1)
R is a reduced ring; 
(2)
D _{3}(R )is an IFP ring; 
(3)
D _{3}(R )is a regularIFP ting; 
(4)
D _{3}(R )is a unitIFP ring; 
(5)
AN (D _{3}(R ))B = 0whenever AB = 0for A ,B ∈D _{3}(R ).
The equivalences of the conditions (1), (2), and (4) are proved by [15, Proposition 2.1], and so they are equivalent to (3).
(4) ⇒ (5): Suppose that (4) holds and let
(5) ⇒ (1): Suppose that (5) holds. Assume on the contrary that there exists 0 ≠
in
Following Cohn [6], a ring
Example 3.2
We refer to the construction and argument in [16, Example 2.1]. Let
be the free algebra generated by noncommuting indeterminates
where the constant terms of
with their images in
Now, consider
Then
because
Remark 3.3

(1) Note that
D _{2}(R ) over a reduced ringR is IFP by [16, Proposition 1.6] and so it is regularIFP. Moreover, there exists a nonreduced noncommutative reversible ringR over whichD _{2}(R ) is regularIFP by [15, Example 2.2]. However, the ringS is always regularIFP whenD _{2}(S ) is regularIFP. For, suppose thatD _{2}(S ) is regularIFP and letab = 0 fora ,b ∈S . For$A=\left(\begin{array}{cc}a& 0\\ 0& a\end{array}\right),B=\left(\begin{array}{cc}b& 0\\ 0& b\end{array}\right)\in {D}_{2}(S)$ , we haveAB = 0 and soAC (D _{2}(S ))B = 0 by assumption. Set$C=\left(\begin{array}{cc}c& 0\\ 0& c\end{array}\right)$ for anyc ∈C (S ). ThenC ∈C (D _{2}(S )) andACB = 0, entailingacb = 0. ThusS is regularIFP. 
(2) Related to (1) above, there exists a reversible ring
R such thatD _{2}(R ) is not regularIFP. Let ℍ be the Hamilton quaternions over ℝ andR =D _{2}(ℍ). ThenR is reversible [16, Proposition 1.6]. We refer to the argument in [16, Example 1.7]. Consider$$A=\left(\begin{array}{cc}\left(\begin{array}{cc}0& i\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}j& 0\\ 0& j\end{array}\right)\\ \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}0& i\\ 0& 0\end{array}\right)\end{array}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and\hspace{0.17em}}B=\left(\begin{array}{cc}\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}k& 0\\ 0& k\end{array}\right)\\ \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\end{array}\right)$$ in
D _{2}(R ). ThenAB = 0.Note that
$$C=\left(\begin{array}{cc}\left(\begin{array}{cc}j& 0\\ 0& j\end{array}\right)& \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\\ \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}j& 0\\ 0& j\end{array}\right)\end{array}\right)\in C({D}_{2}(R))$$ by [13, Lemma 2.1] because
$\left(\begin{array}{cc}j& 0\\ 0& j\end{array}\right)\in C({D}_{2}(\mathbb{H}))$ . ButACB ≠ 0, henceD _{2}(R ) is not regularIFP. 
(3) For a ring
R andn ≥ 2, letV _{n} (R ) be the ring of all matrices (a _{ij} ) inD _{n} (R ) such thata _{st} =a _{(}_{s} _{+1)(}_{t} _{+1)} fors = 1, …,n − 2 andt = 2, …,n − 1. Note that${V}_{n}(R)\cong {\scriptstyle \frac{R[x]}{{x}^{n}R[x]}}$ . IfR is a reduced ring, thenV _{n} (R ) is (regular)IFP by [17, Lemma 2.3 and Proposition 3.3], but the converse does not hold in general as can be seen by the commutative ringV _{n} (R ) over a nonreduced commutative ring (e.g., ℤ_{n} _{l} forn ,l ≥ 2)R forn ≥ 2.
Proposition 3.4

Let M be a multiplicatively closed subset of a ring R consisting of central regular elements. Then R is regularIFP if and only if M ^{−1}R is regularIFP . 
Let R be a ring. Then R [x ]is regularIFP if and only if R [x ,x ^{−1}]is regularIFP .
(1) It comes from the fact that
(2) Recall the ring of
In [15, Example 2.7], we see an IFP ring
Proposition 3.5

{
c +c _{1}x + … +c _{t} x ^{t} ∈R [x ] c ∈C (R )for t ≥ 1} ⊆C (R [x ])and {d _{0} +d _{1}x + … +d _{s} _{−1}x ^{s} ^{−1} +dx ^{s} ∈R [x ] d ∈C (R )for s ≥ 1} ⊆C (R [x ]). 
If R [x ]is regularIFP, then so is R . 
Let R be a regularIFP ring such that C (R [x ]) = {c +xN (R )[x ] c ∈C (R )}. If R is Armendariz then R [x ]is regularIFP .
(1) Consider
(2) It is routine.
(3) Suppose that
The next example shows that the condition “
Example 3.6
We use the ring and argument in [12, Example 2]. Let
where
Notice that (
because
Considering Proposition 3.5, it is natural to ask whether
Example 3.7
Let
This implies
We consider next some equivalent conditions to the regularIFP property in relation to the sum of coefficients of polynomials which satisfy some property of inserting regular polynomials. For
Proposition 3.8

(1)
R is regularIFP; 
(2)
If f _{1}(x )f _{2}(x ) ⋯f _{n} (x ) = 0for f _{1}(x ),f _{2}(x ), …,f _{n} (x ) ∈R [x ],then the sum of all coefficients of every polynomial in $${f}_{1}(x)C(R)[x]{f}_{2}(x)C(R)[x]\cdots {f}_{n1}(x)C(R)[x]{f}_{n}(x)$$ is zero; 
(3)
If f (x )g (x ) = 0for f (x ),g (x ) ∈R [x ],then the sum of all coefficients of every polynomial in f (x )C (R )[x ]g (x )is zero; 
(4)
If f (x )g (x ) = 0for linear polynomials f (x ),g (x )in R [x ],then the sum of all coefficients of every polynomial in f (x )C (R )[x ]g (x )is zero; 
(5)
f (x )g (x ) = 0implies f (x )C (R )[x ]g (x ) = 0for linear polynomials f (x ),g (x )in R [x ].
The procedure of the proof is almost similar to one of [15, Proposition 2.8], but we write it here for completeness. (1) ⇒ (2): Assume that the condition (1) holds. Let
By Remark 2.3(1), we have
is zero.
(2) ⇒ (3), (3) ⇒ (4), and (5) ⇒ (1) are obvious.
(4) ⇒ (5): Assume that the condition (4) holds. Let
From
for all
4. Related Topic
Based on Proposition 3.1(5), a ring
Theorem 4.1

(1)
N _{0}(R ) =N _{*}(R ) =N ^{*}(R ). 
(2)
J (R [x ]) =N _{0}(R [x ]) =N _{*}(R [x ]) =N ^{*}(R [x ]) =N _{0}(R )[x ] =N _{*}(R )[x ] =N ^{*}(R )[x ] =N _{0}(R )[x ] ⊆J (R )[x ]. Moreover, J (R [x ]) =J (R )[x ]when J (R )is nil .
(1) Let
and hence (
(2) By help of [1, Theorem 1] and [5, Corollary 4], we have
and so
Moreover,
Notice that
On the other hand, Hong et al. [9] consider the duo property on the monoid of regular elements as follows. They call a ring
Proposition 4.2
Suppose that
Notice that the converse of Proposition 4.2 does not hold in general by the next example.
Example 4.3
Let
Consider the group ring
Corollary 4.4

(1)
R is DR; 
(2)
R is regularIFP; 
(3)
R is unitIFP; 
(4)
R is nilpotentIFP; 
(5)
R is Abelian; 
(6)
The equation 1 +x ^{2} +y ^{2} = 0has no solutions in K; 
(7)
R is isomorphic to a finite direct product of division rings; 
(8)
R is reduced .
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