A Generalized Higher Reverse Left (accordingly, Right) Centralizer on Prime Rings

第 54 卷 第 5 期

2019 年 10 月

JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY

Vol.54 No.5

Oct. 2019

ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.54.5.47

A

G

ENERALIZED

H

IGHER

R

EVERSE

L

EFT

(

ACCORDINGLY

,

R

IGHT

)

C

ENTRALIZER ON

P

RIME

R

INGS

Fawaz Ra’ad Jarullah,a

Salah Mehdi Salih b

a

Department of Mathematics , College of Education , Al-Mustansirya University , Iraq [email protected]

b _{Department of Mathematics , College of Education , Al-Mustansirya University , Iraq}

[email protected]

Abstract

In this study we introduced the concepts of generalized higher reverse left (accordingly, right) centralizer, and Jordan generalized higher reverse left (accordingly, right) centralizer of rings. The definition of Jordan triple generalized higher reverse left (accordingly, right) centralizer was deduced. The most important findings of this paper are as follows: every Jordan generalized higher reverse left (accordingly right) centralizer of a 2-torsion free prime ring R into itself is a generalized higher reverse left (accordingly right) centralizer of R. The results have confirmed that every Jordan generalized higher reverse left (accordingly right) centralizer is a generalized higher reverse left (accordingly right) centralizer within certain conditions.

Keywords: prime ring, left centralizer, right centralizer, Jordan centralizer

摘要 : 在这项研究中，我们介绍了广义上偏左偏左（相应地，右）定心器的概念，而约旦广义上偏左偏左（相应 地，右）定心器。推导了约旦三元广义广义左上定心（相应地右定心）的定义。 本文最重要的发现如下： 每个约旦将 2 扭转自由质环 R 的左上偏左偏心（相应地右）定心器是 R 的左上偏偏左偏心（相应地右）定 心器。结果已确认，在某些条件下，每个 Jordan 广义左上偏左（相应地右）扶正器都是广义左上偏右（相 应地右）扶正器。 关键词: 填料环，左扶正器，右扶正器，乔丹扶正器

I. I

NTRODUCTION

The concept of generalized higher reverse left (accordingly right) centralizer is the one of most important topics in non-commutative algebra. The objective of the definition of generalized higher reverse left (accordingly right) centralizer is to generalize the definition of higher reverse left (accordingly right) centralizer on rings and to specify the relation between two concepts: generalized higher reverse left (accordingly right) centralizer and Jordan generalized higher reverse left (accordingly right) centralizer within certain conditions.

The definition of prime ring and semi-prime ring was introduced in [1]. The definition of

2-torsion free ring was introduced in [2]. While [3] introduced the concept of left (accordingly, right) centralizer and Jordan left (accordingly, right) centralizer of rings, and thus proved that any left (accordingly, right) Jordan centralizer of a 2-torsion free semi-prime ring is a left (accordingly, right) centralizer. On the other hand, [3] was first to introduce the concepts of higher left centralizer, Jordan higher left centralizer and generalization on rings, many results were found by the researchers, one of these results was that every Jordan higher left centralizer of a 2-torsion free prime ring R is a higher left centralizer.

The concept of generalized higher reverse left (accordingly, right) centralizer of -rings was

first introduced in [4]. One important question can be answered in this paper whether there is a relation between a concept of Jordan generalized higher reverse left(accordingly right) centralizer and a concept of Jordan triple generalized higher reverse left (accordingly right) centralizer within certain conditions.

II. G

ENERALIZED

H

IGHER

R

EVERSE

L

EFT

(A

CCORDINGLY

,

R

IGHT

)

C

ENTRALIZER ON

P

RIME

R

INGS Definition 2.1

Let T = (Ti)iN be a family of additive

mappings of a ring R into itself. Then T is called a generalized higher reverse left (accordingly right) centralizer of a ring R into itself associated with the higher reverse left (accordingly right) centralizer t = (ti) i N of R if for all x, y  R and

n  N: n n i i - 1 i 1 T (x y) T (y) t (x)  



(1) (accordingly, n n i - 1 i i 1 T (x y) t (y) T (x)  



). Definition 2.2

Let T = (Ti) iN be a family of additive

mappings of a ring R into itself. Then T is called a Jordan generalized higher reverse left (accordingly right) centralizer of a ring R into

itself associated with the Jordan higher reverse left (accordingly right) centralizer t = (ti) i N of R

if for all x  R and n  N:

n n i i -1 2 1 i T (x ) T (x) t (x)  



(2) (accordingly, n n i - 1 i i 1 2 T (x ) t (x) T (x)  



) Definition 2.3

Let T = (Ti)iN be a family of additive

mappings of a ring R into itself. Then T is called a Jordan triple generalized higher reverse left (accordingly right) centralizer of a ring R into itself associated with the Jordan triple higher reverse left (accordingly right) centralizer t = (ti)iN of R if for all x y  R and n  N:

n n i i - 1 i - 1 i 1 T (xyx) T (x) t (y) t (x)  



(3) (accordingly, _n n i - 1 i - 1 i i 1 T (xyx) t (x) t (y) T (x)  



) Lemma 2.4

Let T = (Ti)i N be a Jordan generalized higher

reverse left (accordingly , right) centralizer of a ring R into itself .Then for all x, y , z  R and n  N, the following equation holds:

(i) n n n i i - 1 i i - 1 i 1 i 1 T (x y y x) T (y) t (x) T (x) t (y)    







(4) (accordingly, n n n i - 1 i i - 1 i i 1 i 1 T (x y y x) t (y)T (x) t (x)T (y)    







)

 

n n i i - 1 i - 1 n i i - 1 i -1 i 1 i 1 T (x y z z y x) T (z) t (y) t (x) T (x) t (y) t (z)    







ii (5) (accordingly, n n n i - 1 i - 1 i i -1 i - 1 i i 1 i 1 T (x y z z y x) t (z) t (y) T (x) t (x) t (y) T (z))    







)

(iii) In particular, if R is a 2-torsion free commutative ring, then:

n n i i - 1 i - 1 i 1 T (x y z) T (z) t (y) t (x)  



(6) (accordingly, n n i - 1 i - 1 i i 1 T (x y z) t (z) t (y) T (x)  



) Proof:

(i) Since t is a Jordan generalized higher reverse left (accordingly right) centralizer, we have:

n n i i - 1 i 1 T ((x y) (x y)) T (x y) t (x y)    



  n n n n i i - 1 i i - 1 i i - 1 i i - 1 i 1 i 1 i 1 i 1

T (x) α t (x) T (x) α t (y) T (y) α t (x) T (y) α t (y)...(1)

   

















Meanwhile,

n n

= T (x α x)n T (y α y)n T (x α y+ y α x)n

n n

i i - 1 i i - 1 n

i 1 i 1

T (x) α t (x) T (y) α t (y) T (x α y yα x)

 









  (2)

We obtain the following equation by comparing equations (1) and (2)

n n n i i - 1 i i - 1 i 1 i 1 T (x y y x) T (y) t (x) T (x) t (y)    







(ii) By substituting (x yy x) for y in Lemma (2.4) (i), we obtain:

n i i - 1 i - 1 i i - 1 i - 1 i 1 i i - 1 i - 1 i i - 1 i - 1 T (y) t (x) t (x) T (x) t (y) t (x) T (x) t (y) t (x) T (x) t (x) t (y)...(1)     



In addition, we obtain that

n T (x ( x y yx) ( xy yx ) x) = n i i - 1 i - 1 i 1 T (y) t (x) t (x) 



+ T (x) ti i - 1(x) ti - 1(y) + T (n xyx xyx). ..(2)

We get the following equation by comparing equations (1) and (2)

n n n i i - 1 i - 1 i i - 1 i -1 i 1 i 1 T (x y z z y x) T (z) t (y) t (x) T (x) t (y) t (z)    







(iii) Using (ii) and since R is a commutative ring, we have

n

T (xyzxyz)



T (2 xyz)

_n n i i-1 i-1 i 1 2 T (z) t (y) t (x)  



Since R is a 2-torsion free ring, we get

n T (xyz) n i i-1 i-1 i 1 T (z) t (y) t (x)  



Definition 2.5

Let T = (Ti)i N be a Jordan generalized higher

reverse left (accordingly right) centralizer of a ring R into itself .Then for all x, y  R, we

define n: RR R by n( x ,y )= n n i i - 1 i 1 T (x y) T (y) t (x)  



(7) (accordingly, n(x ,y) = i - 1 n n i i 1 T (x y) t (y) T (x)  



₎ Lemma 2.6

Let T = (Ti)i N be a Jordan generalized higher

reverse left (accordingly right) centralizer of a ring R ring into itself. Then for all x, y, z  R, we have that the following equations hold:

(i) n(x, y) = – n(x ,y)

(ii) n(x + y , z) = n (x, z) + n(y, z)

(iii) n(x, y + z) = n(x ,y)+ n(x, z)

Proof:

(i) By applying Lemma (2.4) (i), we have that

n n n i i - 1 i i - 1 i 1 i 1 T (x y y x) T (y) t (x) T (x) t (y)    







n n n i i - 1 n i i - 1 i 1 i 1 T (x y) T (y) t (x) (T (y x) T (x) t (y))   



  



Thus, n(x ,y) = – n(x, y) (ii) n(x+y,z)= n n i i - 1 i 1 T ((x + y)z ) T (z)t (x + y)  



n n n i i - 1 i i - 1 i 1 i 1 T (x z y z ) T (z) t (x) T (z) t (y)     







n n n i i - 1 n i i - 1 i 1 i 1 T (x z) T (z) t (x) T (y z ) T (z) t (y)    



 



= n (x, z) + n(y, z) (iii) n(x,y+z)= n n i i - 1 i 1 T (x ( y z )) T (y z) t (x)   



 n n n i i-1 i i-1 i 1 i 1 T (x y x z ) T (y) t (x) T (z) t (x)     







n n n i i-1 n i i-1 i 1 i 1 T (x y) T (y) t (x) T (xz) T (z) t (x)    



 



= n(x, y) + n(x, z).

Remark 2.7 It is noteworthy that T = (Ti) i N is

a generalized higher reverse left (accordingly right) centralizer of a ring R, into itself if and only if n (x, y) = 0, for all x, y  R and n  N.

Let T = (Ti)iN be a Jordan generalized higher

reverse left (accordingly right) centralizer of a prime ring R into itself. Then for all x, y, z  R and n  N, the following equation holds:

n(x ,y) tn -1(z) [tn - 1( x ), tn - 1(y)]= 0 (8)

Proof:

(i) The Proof is utilizing induction on n  N If n = 1,

Let w = x y z y x + y x z x y

Then, we obtain that

T(w) = T(x( y z y ) x + y (x z x) y) = T (x) y z y x + T(y) x z x y (9) Moreover, we have that

T(w) = T((xy) z (y x) + (yx) z (x y)) = T(yx) z y x + T(xy)z x y (10) The comparison of equations (9) and (10) yields that

0 = (T(yx) – T(x)y) z y x + (T(xy) – T(y)x) z x y 0 = (y, x) z y x +  (x , y) z x y

0 = – (x, y) z y x + (x , y) z x y 0 = (x, y) z (x y – y x)

Thus, (x, y) z [x, y] = 0, for all x, y, z  R. Now, we assume the following:

s (x, y) ts-1( z) [ts - 1 (x), t s - 1 (y)] = 0, for all x, y, z  R , s, n  N and s < n. (11)

Tn(w) = Tn (x( y z y ) x + y (x z x) y) = n i 1



Ti (x) ti - 1 (y z y) ti -1( x ) + n i 1



Ti(y)ti - 1 (x z x) ti - 1( y ) (12) = n i 1



Ti (x) ti - 1(y) ti - 1(z) ti - 1 (y) ti - 1(x) + n i 1



Ti (y) ti - 1(x) ti - 1(z) ti - 1 (x) ti - 1(y) (13) = n i 1 ( 



Ti(x) ti – 1(y) ) tn - 1(z) tn - 1(y) tn - 1(x) + n 1 i 1  



Ti(x)ti - 1 (y) ti - 1(z) ti - 1(y) ti - 1(x) + n i 1 ( 



Ti(y) ti - 1 (x) ) tn - 1(z) tn - 1(x) tn - 1( y) + n 1 i 1  



Ti(y) ti - 1(x) ti - 1 (z) ti – 1(x) ti - 1(y) (14)

Tn(w) = Tn ((xy) z (y x) + (yx) z (xy)) = n i 1



Ti (yx) ti - 1 (z) ti - 1 (x y) + n i 1



Ti (xy) ti - 1 (z) ti - 1 (y x) = Tn (yx) tn - 1 (z) tn - 1(y) tn - 1(x) + n 1 i 1  



Tn (yx) ti - 1 (z) ti – 1(y) ti - 1 (x) + Tn (xy) tn - 1 (z) tn - 1 (x) tn - 1 (y) + n 1

i 1

 



Tn(xy) ti - 1 (z) ti - 1 (x) ti - 1 (y) (15 )

By comparing equations (14) and (15), we have that 0 = (Tn (yx)

n i 1



Ti(x) ti - 1 (y)) tn - 1(z) tn - 1 (y) tn - 1(x) + (Tn (xy) n i 1



Ti(y) ti - 1 (x)) tn - 1(z) tn - 1 (x) tn - 1 (y) + n 1 i 1  



(Ti (yx) Ti (x) ti - 1 (y)) ti - 1(z) ti - 1 (y) ti - 1 (x) + n 1 i 1  



(Ti (xy) Ti(y) t i - 1 (x)) t i - 1(z) t i - 1 (x) t i - 1 (y) (16) It follows that









0 = n(y, x) t n - 1 (z) t n-1 (y) t n - 1 (x) + n(x ,y) t n -1 (z) t n - 1 (x) t n - 1 (y)

0 = n(x, y) t n - 1 (z) t n - 1 (y)t n - 1 (x) + n(x, y) t n - 1 (z) t n - 1 (x) t n - 1 (y)

0 = n(x, y) t n - 1 (z)(t n - 1(x) t n - 1 (y) t n - 1 (y) t n – 1 (x))

Thus, n (x, y) t n - 1 (z) [t n - 1 (x), t n - 1 (y)] = 0, for all x, y, z  R and n  N

Lemma 2.9

Let T = (Ti)i N be a Jordan generalized higher

reverse left (accordingly right) centralizer of a

prime ring R into itself. Then for all x, y, z, u, v  R and n  N:

n(x ,y) tn -1(z)[tn - 1 (u), tn - 1 (v)] = 0 (17)

Proof:

(i) By substituting (x + u) for x in Lemma 2.8, we have that n(x + u ,y) tn -1(z) [ tn - 1(x + u) , tn - 1(y) ] = 0.

Thus

n(x, y) tn -1(z)[tn - 1(x), tn - 1(y)] + n(x ,y) tn -1(z) [tn - 1(u) , tn - 1(y)]

n(u, y) tn -1(z) [tn - 1(x), tn - 1(y)]+ n(u, y) tn -1(z) [tn - 1(u), tn - 1(y)] = 0.

By applying Lemma 2.8, we obtain that

n(x, y) tn -1(z) [tn - 1(u), tn - 1(y)] + n(u, y) tn -1(z) [tn - 1(x), tn - 1(y)] = 0

Therefore, we get that

n(x, y) tn -1(z) [tn - 1(u), tn - 1(y)] tn -1(z) n(x, y) tn -1(z) [tn - 1(u) ,tn - 1(y)] = 0.

This implies that

0 = n(x, y) tn -1(z) [tn - 1(u), tn - 1(y)] tn -1(z) n(u, y) tn -1(z) [tn - 1(x), tn - 1(y)] .

Since R is a prime ring, we have that

n(x, y) tn -1(z) [tn - 1(u), tn - 1(y)] = 0 (18)

The substitution of (y + v) for y in Lemma 2.8, gives that n(x, y + v) tn -1(z) [tn - 1(x), tn - 1(y + v)] = 0

n(x, y) tn -1(z) [tn - 1(x) , tn - 1(y)] + n(x, y) tn -1(z) [tn - 1(x) , tn - 1(v)] + n(x, v) tn -1(z) [tn - 1(x), tn - 1(y)] +

n(x ,v) tn -1(z) [tn - 1(x), tn - 1(v)] = 0.

By utilizing Lemma 2.8, we obtain that

n(x ,y) tn -1(z) [ tn - 1(x) ,tn - 1(v) ] + n(x ,y) tn -1(z) [tn - 1(x),tn - 1(y)] = 0

Consequently, we have

n(x, y) tn -1(z)[tn - 1(x) ,tn - 1(v)] tn -1(z)n(x, y) tn -1(z) [tn - 1(x) ,tn - 1(v)] = 0

This implies that

0 = n(x, y) tn -1(z) [tn - 1(x), tn - 1(v)] tn -1(z) n(x ,v) tn -1(z) [tn - 1(x), tn - 1(y)].

The fact that R is a prime number yields that

n(x,y) tn -1(z)[tn - 1(x) ,tn - 1(v)] = 0 (19)

Now, n(x ,y) tn -1(z) [ tn - 1(x + u) ,tn - 1( y+v) ] = 0

n(x, y) tn -1(z) [ tn - 1(x) ,tn - 1(y) ] + n(x ,y) tn -1(z) [ tn - 1(x) ,tn - 1(v) ] + n(x ,y) tn -1(z) [ tn - 1(u), tn - 1(y)]+

n(x, y) tn -1(z)[tn - 1(u), tn - 1(v)] = 0.







第 54 卷 第 5 期

2019 年 10 月

JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY

Vol.54 No.5

Oct. 2019

By employing equations (18), (19) and Lemma 2.8, we get

n(x, y) tn -1(z) [tn - 1(u), tn - 1(v)] = 0.

Theorem 2.10

Every Jordan generalized higher reverse left (accordingly right) centralizer of a 2-torsion free prime ring R into itself is a generalized higher reverse left (accordingly right) centralizer of R. Proof:

Let T = (Ti)i N be a Jordan generalized higher

reverse left (accordingly right) centralizer of a prime ring R into itself. Since R is a prime ring, then by employing Lemma 2.9, we have that either n(x, y) = 0 or [tn - 1(u), tn - 1(v)] = 0, for all

x, y, u, v  R and n  N.

If [tn – 1(u), tn - 1(v)]  0, for all u, v  R, then

n(x, y) = 0, for all x, y  R and n  N. Hence,

using Remark 2.7, we obtain that T is a higher

generalized reverse left (accordingly right) centralizer of R.

If [tn - 1(u), tn - 1(v)] = 0, for all u, v  R and n  N.

Then R is a commutative ring. By utilizing Lemma 2.4(i), we have that

n n i i - 1 i 1 T (x y x y ) 2 T (y) t (x)   



(20) Since R is a 2-torsion free ring, we get that

n n i i-1 i 1 T (x y) T (y) t (x)  



(21) Then T is a higher generalized reverse left (accordingly right) centralizer of R.

Proposition 2.11

Let T = (Ti)i N be a Jordan generalized higher

reverse left (accordingly right) centralizer of a 2-torsion free ring R into itself. Then T is a

Jordan triple generalized higher reverse left (accordingly right) centralizer of R.

Proof:

The substitution of y for (x y + y x) in Lemma 2.4 (i) results in



 



n x y + yx x y + T (x  yx x)

 

n i i - 1 i - 1 i i - 1 i - 1 i 1 i i - 1 i - 1 i i - 1 i - 1 T (y) t (x) t (x) T (x) t (y) t (x) T (x) t (y) t (x) T (x) t (x) t (y) 23     



Moreover, we get that

n T (x( xyyx) ( xyy )x)x = n 2 2 T (x yxy x x y xy x ) = n i i - 1 i - 1 i 1 T (y) t (x) t (x) 



+ T (x) ti i - 1(x) ti - 1(y) + T ( y x x y xn x  )(24)

By comparing equations (23) and (24)

n T ( y x x x y x) ₌ n x 2 T ( y x) = n i i - 1 i - 1 i 1 2 T (x) t (y) t (x) 



(25)

Since R is a 2-torsion free ring, we obtain that T is a Jordan triple generalized higher reverse left (accordingly right) centralizer of R.

III. C

ONCLUSION

In this paper we defined the concepts of generalized higher reverse left (accordingly right) centralizer, Jordan generalized higher reverse left (accordingly right) centralizer and Jordan triple generalized higher reverse left (accordingly right) centralizer on rings. This paper discussed the relation between the concept of Jordan generalized higher reverse left (accordingly right) centralizer and Jordan generalized higher reverse left (accordingly right) centralizer within certain conditions. Also, this paper was able to answer the question that referred to the introduction.

According to the previous results, the future study in this paper will indicate the domain of additive mapping that deals with a M-Module while the Codomain deals with a -ring. So that we can give the definition of the concept of generalized higher reverse left (accordingly right) centralizer and we can find many results with fewer conditions. Through a future study we will be able to discover a relation between a commutative algebra represented by a  M-Module and a non-commutative algebra represented by -ring.

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