# An Equivalent Condition and Some Properties of StrongJ-Symmetric Ring

Authors: Shun Xu
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetJR denote the Jacobson radical of a ring R. We say that ring R is strong J-symmetric if, for any a,b,c∈R, abc∈JR implies bac∈JR. If ring R is strong J-symmetric, then it is proved that Rx/xn is strong J-symmetric for any n≥2. If R and S are rings and WSR is a R,S-bimodule, E=TR,S,W=RW0S=rw0s|r∈R,w∈W,s∈S,then it is proved that R and S are J-symmetric if and only if E is J-symmetric. It is also proved that R and S are strong J-symmetric if and only if E is strong J-symmetric.


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## Body

## 1. Introduction

R is a ring with an identity element. The symbols JR,NR,ZR,andER, respectively, stand for the Jacobson radical, the set of all nilpotent elements, the set of all central elements, and the set of all idempotent elements of R. We also denote the set a∈R:a2=0 by N2R and denote the ring of n×n upper triangular matrix over R by TnR. R is left quasi-duo if every maximal left ideal of R is an ideal.Various generalizations of symmetric rings have been carried out by many authors over the last several years.R is symmetric [1] if abc=0 implies acb=0, for all a,b,c∈R. The study of symmetric rings also can be found in [2]. R is weak symmetric [3] if, for any a,b,c∈R, abc∈NR implies acb∈NR. R is central symmetric [4] if, for any a,b,c∈R, abc=0 implies bac∈ZR. R is generalized weakly symmetric (GWS) [5] if, for any a,b,c∈R, abc=0 implies bac∈NR. It follows that the class of GWS rings contains the class of weak symmetric rings. Moreover, it is known that central symmetric rings are GWS [5]. Let e be an idempotent of R; R is called an e-symmetric ring [6] if abc=0 implies acbe=0, for all a,b,c∈R. And, Wei gives some properties of e-symmetric rings in [7].As a generalization of symmetric rings, Kwak at el. [8] extended the concept of symmetric rings to ideal-symmetric rings. A ring R is called ideal symmetric if ABC=0 implies ACB=0 for all ideals A,B,C of R. Chakraborty and Das called a ring R right (respectively, left) nil-symmetric [9] if whenever abc=0 (respectively, cab=0), for every a,b∈NRand every c∈R, then acb=0, and the ring R is nil-symmetric if it is both right and left nil-symmetric. In [10], a ring R is called weak right (respectively, left) nil-symmetric if abc=0, for all a,b,c∈NR implies acb=0 (respectively, bac=0), and the ring R is called weak nil-symmetric if it is both weak left nil-symmetric and weak right nil-symmetric. A ring R will be said to be right (respectively, left) nil-ideal symmetric [11] if IJK=0 implies IKJ=0 (respectively, JIK=0) for nil ideals I,J,K of R. Lambek called a right ideal I of a ring R symmetric if rst∈I implies rts∈I, for all rst∈R. If the zero ideal is symmetric then R is usually called symmetric; while Anderson and Camillo [12] used the term ZC3 for this concept, Han call an ideal N of a ring R ideal symmetric [13] if ABC⊆N implies ACB⊆N, for any ideals A,B,C in R. Note that if the zero ideal of a ring R is ideal symmetric, then R is ideal symmetric.In this paper, we introduce a kind of new symmetrical ring, which is strongJ-symmetric. A ring R is strong J-symmetric if, for any a,b,c∈R, abc=0 implies bac∈JR.


## 2. Main Results

Definition 1.
A ringR is J-symmetric if, for any a,b,c∈R, abc=0 implies bac∈JR.Definition 2.
A ringR is strong J-symmetric if, for any a,b,c∈R, abc∈JR implies bac∈JR
LetR be a ring. For any a,b,c∈R, if abc=0, then we have 0=abc∈JR. If a ring R is strong J-symmetric, then we have bac∈JR. Therefore, R is J-symmetric. We obtain(1)ring⋅R|⋅R⋅is⋅strong⋅J−symmetric⋅⊑⋅ring⋅R|⋅R⋅is⋅J−symmetric.
Therefore, strongJ-symmetric ring is “strong.”Definition 3.
A ringR is strong central symmetric if, for any a,b,c∈R, abc∈JR implies bac∈ZR.Definition 4.
A ringR is strong generalized weakly symmetric if, for any a,b,c∈R, abc∈JR implies bac∈NR.
The following lemma [14] is well known.Lemma 1.
IfR is a ring and x∈R, then x∈JR if and only if, for any y∈R, there exists u∈R such that 1−xyu=1 and if and only if, for any y∈R, there exists v∈R such that v1−yx=1.The above lemma plays a critical role in proving the properties of strongJ-symmetric ring because we use it repeatedly. Next, we introduce three basic properties of strong J-symmetric ring.Proposition 1.
For any ringR, the following statements are equivalent:(1)
R is strong J-symmetric(2)
For anya,b,c∈R, abc∈JR implies acb∈JR.Proof.
1⇒2 Let a,b,c∈R such that abc∈JR. If acb∉JR, then there is a left maximal ideal M⊆R such that acb∉M. Therefore, we have M+Racb=R. There exists x∈M and y∈R such that x+yacb=1. Since abc∈JR, we have yabc∈JR. Since JR is strong J-symmetric, we have byac∈JR. Since 1−x2=yacbyacb∈JR⊆M, we have x∈M. Therefore, 1∈M, that is a contradiction.2⇒1 Let a,b,c∈R such that abc∈JR. If bac∉JR, then there is left maximal ideal M⊆R such that bac∉M. Therefore, we have M+Rbac=R. There exists x∈M,y∈R such that x+ybac=1. Since abc∈JR, we have abcy∈JR. We have acyb∈JR by (2). Since 1−x2=ybacybac∈JR⊆M, we have x∈M. Therefore, 1∈M, that is a contradiction.Corollary 1.
LetR be a strong J-symmetric ring. For any a,b,c∈R, if abc∈JR, then acb,bac,bca,cab,cba∈JRProposition 2.
LetR be a strong J-symmetric ring. For any left maximal ideal M of R and any a,b,c∈R, if abc∈JR, then a∈M or bc∈M.Proof.
For any left maximal idealM of R, if a∉M, then M+Ra=R. There exist x∈Mandy∈R such that x+ya=1. We get x−1bc∈JR⊆M. Since abc∈JR, we have bc∈M.
LetR be a strong J-symmetric ring. For any e∈ER and any left maximal ideal M of R, since 0=e2−e=e1−e=e1−e⋅1, we get e∈M or 1−e∈M from Proposition 2.
For anya∈N2R, we have 0=a2=a⋅a⋅1. We get a∈JR from Proposition 2. Therefore, N2R⊆JR.Proposition 3.
LetR be a left quasi-duo ring. For any a,b,c∈R, if abc∈JR, then a∈M or b∈M or c∈M for any maximal left ideal M of R.Proof.
LetM be a maximal left ideal of R. If a∉M, then M+Ra=R, which implies that x+ya=1, for some x∈M,y∈R. Therefore, xbc=bc−yabc. When R is left quasi-duo and x∈M, we get bc−yabc∈M. Since yabc∈JR⊆M, we have bc∈M. If b∉M, then M+Rb=R yields u+vb=1, for some u∈M,v∈R. Since 1−vb∈M, we have 1−vbc∈M. Therefore, we obtain c∈M by using bc∈M.
We want to know which types of rings are strongJ-symmetric rings. From the following three propositions, we can see that left quasi-duo ring, strong central symmetric ring, and strong generalized weakly symmetric ring are strong J-symmetric rings.Proposition 4.
A left quasi-duo ring is strongJ-symmetric.Proof.
LetR be a left quasi-duo ring and abc∈JR and M be a maximal left ideal of R. It follows from Proposition 3 that a∈M or b∈M or c∈M. As R is left quasi-duo, we get bac∈M. Therefore, bac∈JR, which proves that R is strong J-symmetric.Proposition 5.
Strong central symmetric rings are strongJ-symmetric.Proof.
LetR be a strong central symmetric ring which is not strong J-symmetric. There exists a,b,c∈R such that abc∈JR, but bac∉JR so that there exists a maximal left ideal M⊆R such that 1=x+ybac, for some x∈M and y∈R. Now, for any r1,r2∈R, we have abcr11∈JR and r2abc∈JR. Hence, cr1ab and br2ac∈ZR. Therefore, we have(2)1−x4=ybac4=ybacybacybacybac=ybacybabaccyybac=ybacybabaccyybac=ybacybabaccbacyy=ybacybabaccbacyy=ycbabacybabaccyy=ycbabacybabaccyy=ycbabcybabaaccyy=ycbabcybabaaccyy=0.
This leads to1∈M, which is a contradiction. Hence, R is strong J-symmetric.Proposition 6.
Strong generalized weakly symmetric rings are strongJ-symmetricProof.
LetR be a strong generalized weakly symmetric ring and abc∈JR. If R is not strong J-symmetric, then there exists a maximal left ideal M of R such that 1=x+ybac, for some x∈Mandy∈R. As R is strong generalized weakly symmetric and abcy∈JR so that bacy∈NR. We have bacyk=0 for some positive integer k. Therefore, we have(3)1−xk+1=yback+1=ybacykbac=0∈M.
This together withx∈M implies that 1∈M, which is a contradiction. Hence, R is strong J-symmetric.
We want to know if a ring is a strongJ-symmetric ring. From the next three propositions, we can know whether a ring is strong J-symmetric by the structure of quotient ring.Proposition 7.
IfR/JR is symmetric, then R is strong J-symmetric.Proof.
Leta,b,c∈R such that abc∈JR, so a¯b¯c¯=0¯. As R/JR is symmetric, so b¯ac¯=0¯ which means that bac∈JR. Therefore, R is strong J-symmetric.Lemma 2.
LetR be a ring and I be an ideal of R; then, I+JR/I⊆JR/I.Proof.
JR/I=∩I⊑mm/I, where m is maximal left ideal of R, I⊆∩I⊑mm, and JR=∩mm,m is maximal left ideal of R, so JR⊆∩I⊑mm. Therefore, we have I+JR/I⊆JR/I.Proposition 8.
IfI is a nil ideal of a ring R such that R/I is strong J-symmetric, then R is strong J-symmetric.Proof.
Leta,b,c∈R such that abc∈JR. From Lemma 2, we have abc¯∈JR/I in R/I. Since R/I is strong J-symmetric, we have bac¯∈JR/I. For any r∈R, there exists t∈R such that 1−t1−rbac∈I⊆JR from Lemma 1. Let x=1−t1−rbac; since I is nil, there exists positive integer n such that xn=0. Therefore, we have 1+x+⋯+xn−11−x=1. It follows that 1−rbac is left invertible, and hence, bac∈JR. Therefore, R is strong J-symmetricProposition 9.
IfI is an ideal of a J-symmetric ring, S and R are subrings of S containing I and JR=JS; then, R/I is strong J-symmetric, which implies R is strong J-symmetric.Proof.
Leta,b,c∈R such that abc∈JR in R⊆S. Since S is strong J-symmetric and abc∈JS=JR, we have bac∈JS. For any r∈R⊆S, there exists s∈S such that s1−rbac=1. From Lemma 2, we get abc¯∈JR/I. Since R/I is J-symmetric, we have bac¯∈JR/I. Therefore, there exists t∈R such that 1−1−rbact∈I. This yields s−s1−rbact∈I, so s−t∈I⊆R. This implies s∈R. Hence, 1−rbac is left invertible in R so that bac∈JR. Therefore, R is strong J-symmetric.
Next, we consider the direct product andJ-subdirect product of strong J-symmetric rings. From the next two propositions, we can see that the direct product and J-subdirect product of arbitrary family of strong J-symmetric rings are both strong J-symmetric.Proposition 10.
Direct product of arbitrary family of strongJ-symmetric rings is strong J-symmetric.Proof.
For any arbitrary family of ringsRi:i∈I, we know that J∏i∈IRi=∏i∈IJRi. Hence, the result easily follows.Definition 5.
LetRii∈I be a family of rings and R=∏i∈IRi be the direct product of rings Ri with the projections ρi:R⟶Ri. A ring B is called J-subdirect product of Ri if there is an imbedding m:B⟶R such that the homomorphisms ρimi∈I are surjective and JR=JB.Proposition 11.
J-subdirect product of arbitrary family of strong J-symmetric rings is strong J-symmetric.Proof.
LetR be a J-subdirect product of a family of strong J-symmetric rings Rii∈I. For each i∈I, we have epimorphism ϕi:R⟶Ri. Hence, ∏i∈IR/Kerϕi≃∏i∈IRi is strong J-symmetric from Proposition 10. Since the map(4)Φ:R⟶∏i∈IR/Kerϕi,Φr=r+Kerϕii∈I,is a monomorphism, we have R≅ImΦImΦ/ΦKerϕi≃R/Kerϕi which is also a strong J-symmetric. Now, ΦKerϕi⊆ImΦ⊆∏i∈IR/Kerϕi. Therefore, according to Proposition 9, ImΦ≅R is a strong J-symmetric.Theorem 1.
IfR is a strong J-symmetric ring, then(1)
TnR is a strong J-symmetric, for any n≥2.(2)
Rx/xn is a strong J-symmetric, for any n≥2.(3)(5)SnR=aa12…a1n0a…a2n⋮⋮⋱⋮00…a:a,aij∈R,1≤i<j≤n,is a strong J-symmetric, for any n≥2.Proof.
Let(6)I=0a12a13…a1n00a23…a2n⋮⋮⋮⋱⋮⋮00……0:aij∈R,1≤i<j≤n.
Hence,I is a nil ideal of TnR as well as SnR.(1)
TnR/I is isomorphic to direct product of n-copies of R. Therefore, according to Proposition 10 and Proposition 8, TnR is a strong J-symmetric.(2)
Let(7)M=010…0001…⋮⋮⋮⋮⋱⋱100……0n×n,RM=a0In+a1M+⋯+an−1Mn−1|ai∈R,i=0,…,n−1.Consider the map(8)φ:Rx/xn⟶RM,x¯⟶M.We getRx/xn≃RM. Consider(9)VnR=a0a1a2…an−2an−10a0a1…an−2⋮⋮⋮⋮⋮0000…a0:ai∈R,i=0,1,2,…n−1.Therefore, we haveRM=VnR and Rx/xn≃VnR. Consider(10)K=0a1a2…an−2an−100a1…an−2⋮⋮⋮⋮⋮0000…0:ai∈R,i=1,2,⋯n−1.Hence,K is a nil ideal of VnR, and VnR/K≃R is a strong J-symmetric. Therefore, Rx/xn≃VnR is a strong J-symmetric by Proposition 8.(3)
SinceSnR/I≃R, it follows that SnR is also strong J-symmetric by Proposition 8.
LetR and S be rings and WSR be a R,S-bimodule. Let(11)E=TR,S,W=RW0S=rw0s|r∈R,w∈W,s∈S.
For anyr1w10s1,r2w20s2∈E, we define(12)r1w10s1+r2w20s2=r1+r2w2+w10s1+s2,r1w10s1r2w20s2=r1r2r1w2+w1s20s1s2.
Therefore,E is a ring.
WhetherR and S are strong J-symmetric is closely related to whether E is a strong J-symmetric. Exactly, we know that R and S are J-symmetric if and only if E is a J-symmetric from Theorem 2. We also know that R and S are strong J-symmetric if and only if E is a strong J-symmetric from Theorem 3.Lemma 3.
For anyac0b∈E, we have ac0b∈JE if and only if a∈JR and b∈JS.Proof.
⇒ For any r∈R and s∈S, we have r00s∈E. Since ac0b∈JE and uw0v∈E, we have(13)uw0v1R001S−r00sac0b=1R001S.
So,u1−ra=1Randv1−sb=1S. Therefore, we have a∈JRandb∈JS by Lemma 1.
⇐ For any rw0s∈E, since a∈JRandb∈JS, there exist u∈Randv∈S such that(14)u1R−ra=1R,v1S−sb=1S.
There existsuurc+wbv0v∈E such that(15)uurc+wbv0v1R001S−rw0sac0b=1R001S.
Therefore, we haveac0b∈JE by Lemma 1.Theorem 2.
R and S are J-symmetric if and only if E is J-symmetricProof.
⇒ For any(16)r1w10s1,r2w20s2,r3w30s3∈E,if(17)r1w10s1r2w20s2r3w30s3=r1r2r3r1r2w3+w1s3+w1s2s30s1s2s3=0,then r1r2r3=0,s1s2s3=0. Since R and S are J-symmetric, we have r2r1r3∈JR and s1s1s3∈JS. Therefore, we have(18)r2w20s2r1w10s1r3w30s3=r2r1r3r2r1w3+w1s3+w2s1s30s2s1s3∈JE,by Lemma 3. We get E is J-symmetric by Definition 1.
⇐ For any r1,r2,r3∈R and s1,s2,s3∈S, if r1r2r3=0,s1s2s3=0, then(19)r100s1r200s2r300s3=r1r2r300s1s2s3=0.
SinceE is J-symmetric, we have(20)r200s2r100s1r300s3=r2r1r300s2s1s3∈JE.
Therefore, we haver2r1r3∈JRa nd s2s1s3∈JS by Lemma 3. We get R and S are J-symmetric by Definition 1.Theorem 3.
R and S are strong J-symmetric if and only if E is a strong J-symmetric.Proof.
⇒ For any(21)r1w10s1,r2w20s2,r3w30s3∈E,if(22)r1w10s1r2w20s2r3w30ss=r1r2r3r1r2w3+w1s3+w1s2s30s1s2s3∈JE,then r1r2r3∈JR and s1s2s3∈JS by Lemma 3. Since R and S are strong J-symmetric, we have r2r1r3∈JR and s2s1s3∈JS. Therefore, we have(23)r2w20s2r1w10s1r3w30s3=r2r1r3r2r1w3+w1s3+w2s1s30s2s1s3∈JE,by Lemma 3. E is a strong J-symmetric by Definition 2.
⇐ For any r1,r2,r3∈R and s1,s2,s3∈S, if r1r2r3∈JR and s1s2s3∈JS, then(24)r100s1r200s2r300s3=r1r2r300s1s2s3∈JE,by Lemma 3. Since E is a strong J-symmetric, we have(25)r200s2r100s1r300s3=r2r1r300s2s1s3∈JE.
Therefore, we getr2r1r3∈JR and s2s1s3∈JS by Lemma 3. R and S are strong J-symmetric by Definition 2.


## 3. Conclusion

In this paper, we give some basic properties of strongJ-symmetrical ring. We determine that three types of rings are strong J-symmetric rings, which are left quasi-duo ring, strong central symmetric ring, and strong generalized weakly symmetric ring. Next, we consider the direct product and J-subdirect product of strong J-symmetric rings. Besides, we give certain quotient ring and subring of polynomial ring and matrix ring with R coefficients, which are strong J-symmetric ring. Finally, we give a necessary and sufficient condition for a ring to be a J-symmetric ring and for a ring to be a strong J-symmetric ring. Of course, there are many other properties of strong J-symmetric rings that are worthy of our discussion.



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