Let R be an associative ring , U be a nonzero Lie ideal of R. In this paper , we will present the definition of (U,R) strongly derivation pair (d,g) , then we will get d=0 (resp. g=0 ) under &nbsp;certain conditions on d and g for (U,R) strongly derivation pair (d,g) on semiprime ring . After that we will study prime rings , semiprime rings ,and rings &nbsp;that have a commutator left nonzero divisor with (U,R) strongly derivation pair (d,g) , to obtain the notation of&nbsp; (U,R) derivation 

A. Saed, Ikram

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ISSN 2347-1921                                                           3711 | P a g e                                                           J u n e  1 0 ,  2 0 1 5    ( U,R) – STRONGLY  DERIVATION  PAIRS  ON LIE IDEALS  IN  RINGS Ikram  A. Saed Applied Mathematics , Department of Applied Science University of Technology, Baghdad, Iraq. akraam 1962@yahoo.com ABSTRACT Let R be an associative ring , U be a nonzero Lie ideal of R. In this paper , we will present the definition of (U,R) – strongly derivation pair (d,g) , then we will get d=0 (resp. g=0 ) under  certain conditions on d and g for (U,R) – strongly derivation pair (d,g) on semiprime ring . After that we will study prime rings , semiprime rings ,and rings  that have a commutator left nonzero divisor with (U,R) – strongly derivation pair (d,g) , to obtain the notation of  (U,R) – derivation .  Indexing terms/Keywords prime ring; semiprime ring; derivation; strongly derivation pair.                            Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol .10, No.8 www.cirjam.com , editorjam@gmail.com ISSN 2347-1921                                                           3712 | P a g e                                                           J u n e  1 0 ,  2 0 1 5     INTRODUCTION  Let R be an associative ring , U be a nonzero Lie ideal . This paper consists of two section . In section one , we recall some basic definitions and other concepts ,which will be used in our paper , then we explain these concepts by examples ,remarks .In section two , we will present the definition of (U,R) –strongly derivation pair (d,g) and it will be denoted by (U,R) – S- derivation pair (d,g) and we put certain conditions on d and g for (U,R) – strongly derivation pair (d,g) on semiprime ring to obtain d=0 or g=0 . Then we will prove that (1) If  R is a 2- torsion free semiprime ring, and (d,g) be a (U,R) – strongly derivation pair , then d and g are (U,R) – derivations , (2) If  R is a prime ring , and (d,g) be a (U,R) – strongly derivation pair , then d and g are (U,R) – derivations, (3) If  R is a ring which has a commutator left  nonzero divisor and (d,g) be a (U,R) – strongly derivation pair ,then d and g are (U,R) – derivations. 1.BASIC CONCEPTS Definition 1.1:[1] A ring R is called a prime ring if for any a,b R, aRb = {0},implies that either a=0 or b=0 . Examples 1.2:[1]  1.Any integral domain is a prime ring . 2.Any matrix ring over an integral domain is a prime ring . Definition 1.3:[1] A ring R is called a semiprime ring if for any a R,aRa={0},implies that a=0 . Remark 1.4 [1] Every prime ring is a semiprime ring ,but the converse in general is not true. Definition 1.5:[2] A ring R is said to be n-torsion free,where n 0 is an integer if whenever  na=0,with a R , then a=0 . Definition 1.6:[2] A ring R is said to be a commutator right (resp. left ) nonzero divisor ,if there exists elements a and b of R such that  c[a,b]=0 ( resp. [a,b]c=0 ) implies c=0 , for every a R . Definition 1.7:[3] Let R be a ring . Define a lie product [.,.] on R as follows  [x ,y] = xy – yx , for all x,y R. Definition 1.8:[3] An additive subgroup U of a ring R is said to be a Lie ideal of R if the  commutator  [u,r] = ur – ru U , for all  u U , r R . Definition 1.9:[3] A Lie ideal  verifies that  u2 U, for all u U is called  square closed  Lie  ideal . Definition 1.10:[4] Let R be a ring . An additive  mapping d:R  R is called a derivation  if :   d(xy) =d(x)y + xd(y) , for all x,y R . Example 1.11:[4] Let R={ ,a,b N,where N is the ring of integers } be a ring of  2x2  matrices with respect to usual addition and  multiplication . Let  d: R  R , defined by  d  =  , for all  R .Then d  is a derivation  of  R . Definition 1.12:[4] Let U be a Lie ideal of a ring R. An additive  mapping d: R  R is  called  a  (U,R) – derivation ,if  d(xy) = d(x)y + xd(y) ,for all x U ,y R . Example 1.13:[3] Let R= { : x,y,z,w N where N is the ring of integers } be  a ring  of  2x2 matrices with respect to the usual addition and multiplication .                              Let d: R  R , defined by  d  =  , for all   R  and  Let  U= {  , x,y ,z  N } It is clear that  U  is a Lie ideal of R . Then d is a (U,R) – derivation .  ISSN 2347-1921                                                           3713 | P a g e                                                           J u n e  1 0 ,  2 0 1 5     Definition 1.14 :[1] Let R be a ring , additive mappings d,g : R R is called  strongly derivation pair  (d,g) if satisfy the following equations :    d(xy) = d(x)y + xg(y) , for all x,y R .     g(xy) = g(x)y + xd(y) , for all x,y R . 2. (U, R) – STRONGLY  DERIVATION  PAIRS    We first  introduce the basic definition in this section  Definition 2.1: Let R be a ring , U a nonzero Lie ideal of R , additive  mappings    d,g:U  R is called   (U,R) – S – derivation pair (d,g) if satisfy the following equations:       d(xy) = d(x)y + xg(y) , for all x U , y R .         g(xy) = g(x)y + xd(y) , for all x U , y R . Example 2.2: Let R be a non commutative ring , U a nonzero Lie ideal of R and let a,b R , such that  xa = xb = 0 , for all x U . Define d,g : U  R ,as follows :      d(x) = ax  ,  g(x) = bx . Then (d,g) is a (U,R) – S- derivation pair .    Let x U , y R , so :    d(xy) = d(x)y + xg(y) = axy + xby = axy   On the other hand : d(xy) = axy    g(xy) = g(x)y + xd(y) = bxy + xay = bxy   On the other hand : g(xy) = bxy   Hence (d,g) is a  (U,R) – S- derivation pair . Theorem 2.3: Let R be a semiprime ring , U  a nonzero Lie ideal of R with u2 U , for all u U. If R admits a (U,R) – S- derivation pair (d,g), such that d(x)g(y)=0 ( resp. g(x)d(y)=0 ) ,for all x,y U, then d=0 ( resp. g=0 ) . Proof: we have       d(x)g(y) = 0, for all x,y U                                                            (1) Replacing yx for y in (1) and using (1) , we have :      d(x)yd(x)=0 ,for all x,y U                                                              (2) By semiprimeness of R , (2) gives :       d(x)=0 , for all x U                                                                         (3) Similarly , if g(x)d(y)=0 ,for all x,y U ,then g=0 . Theorem 2.4: Let R be a semiprime ring, U a nonzero Lie ideal  of R with u2 U, for all u U. If R admits a (U,R)-S-derivation pair (d,g) , such that d(x) =  (resp.        g(x ) = x ) ,for all x U, then g=0 (resp. d=0 ) . Proof: we  have      d(x) = x , for all x U                                                                          (1) Replacing x by xy in (1) and using (1) , we get :         xg(y) = 0, for all x,y U                                                                  (2) Left multiplication  of (2) by g(y) ,leads to : ISSN 2347-1921                                                           3714 | P a g e                                                           J u n e  1 0 ,  2 0 1 5           g(y)xg(y) = 0 , for all x,y U                                                           (3) By semiprimeness of R , we get :          g(y) = 0 , for all y U                                                              (4) Similarly , we can show if  d(x) = -x , for all x U, then g = 0 In the same way , if g(x) =  , for all x U , then d = 0 . Theorem 2.5 : Let R be a 2- torsion free semiprime ring , U a nonzero Lie ideal of R with u2 U, for all u U, and (d,g) be a (U,R)-S-derivation pair , then d and g are (U,R)-derivations . Proof: Suppose  that (d,g) is (U,R)-S-derivation pair . Then: d(xyx) = d(x(yx))= d(x)yx + xg(yx) , for all x U , y R                      (1) That is : d(xyx) = d(x)yx + xg(y)x + xyd(x) , for all x U , y R                        (2) Also: d(xyx) = d((xy)x) = d(xy)x + xyg(x) , for all x U , y R                      (3) That is : d(xyx) = d(x)yx + xg(y)x + xyg(y) , for all x U , y R                         (4) From (2) and (4) , we get :        xy(d(x) – g(x)) = 0, for all x U , y R                                             (5) Replace y by (d(x) – g(x))yx in (5) , we get :   x(d(x) – g(x)) yx (d(x) – g(x)) = 0, for all x U , y R                         (6) Since R is semiprime , we get :      xd(x) = xg(x) , for all x U                                                                 (7)  It follows that :   d(x2) = d(x)x + xd(x)  , for all x U                                                       (8)  And  g(x2) = g(x)x + xg(x) , for all x U                                                 (9) Thus , by using [1,Theorem 1.1.14] , we obtain that d and g are (U,R)-derivations . Theorem 2.6 : Let R be a prime ring , U a nonzero Lie ideal of R with u2 U, for all u U, and (d,g) be a (U,R)-S-derivation pair , then d and g are (U,R)-derivations . Proof: Since (d,g) is  (U,R)-S-derivation pair , we have (see how relation (5) was obtained from  relation (1) in the proof of Theorem 2.5 )        xy (d(x) – g(x)) = 0 , for all x U , y R                                          (1) And , by primeness of R ,we get :   d(x) = g(x) , for all x U                                                                      (2)  And hence d and g are (U,R)-derivations . Theorem 2.7: Let R be a ring which has a commutator left nonzero divisor , U a nonzero Lie ideal of R with u2 U , for all u U, and (d,g) be a (U,R)-S-derivation pair, then d and g are (U,R)-derivations . ISSN 2347-1921                                                           3715 | P a g e                                                           J u n e  1 0 ,  2 0 1 5     Proof: we have :  d(yx2)  = d(y)x2 + yg(x2) , for all x U, y R                                             (1) That is :  d(yx2) = d(y)x2 + yg(x)x + yxd(x) , for all x U , y R                              (2)   On the other hand :    d(yx2) = d(yx)x  + yxg(x) , for all x U , y R                                         (3)   That is :    d(yx2) = d(y)x2 + yg(x)x + yxg(x) , for all x U ,y R                            (4)   From (2) and (4) , we obtain :    y(xd(x) – xg(x)) = 0 , for all x U, y R                                                  (5)  Replacing y by yr in (5) , to get :    yr(xd(x) – xg(x)) = 0 , for all x U ,y,r  R                                             (6) Again , left multiplying of (5) by r , to get :   ry(xd(x) – xg(x)) = 0 , for all x  U , y ,r R                                            (7)  Subtracting (7)  from (6) , we get :  [y,r] (xd(x) – xg(x)) = 0 ,for all x U , y,r R                                           (8)  Since R has a commutator  left  nonzero divisor , we get :      xd(x) = xg(x) , for all x U                                                                    (9)  Linearizing (9) , we get :  xd(y)  + yd(x) = xg(y) + yg(x) , for all x U , y R                                    (10) That is :  x(d – g)(y) + y(d – g)(x) = 0 , for all x U , y R                         (11)   Replacing y by ry  in (11) , to get :  x(d – g)(ry) + ry(d – g)(x) = 0 , for all x U , y,r R                                 (12) Again , left  multiplying of (11) by r , to get :  rx(d – g)(y) + ry(d – g)(x) = 0 , for all x U , y ,r R                               (13) Subtracting  (12) from (13) , we get :  rx(d – g)(y) – x(d – g)(ry) = 0 , for all x U , y , r R                                (14) Replacing  x by sx in  (14) , to get :  rsx(d – g)(y) – sx(d – g)(ry) = 0 , for all x U , y , r , s R                        (15)  Also , left multiplying of  (14) by  s , to get :  srx(d – g)(y) – sx(d – g)(ry) = 0 , for all x U , y,r,s R                            (16) Subtracting  (16) from (15) ,we get :   [r,s] x (d – g)(y) = 0 , for all x U ,y,r,s R                                               (17) Since R has a commutator  left  nonzero divisor , we get :   x(d – g)(y) = 0, for all x U , y R  .                                                           (18) ISSN 2347-1921                                                           3716 | P a g e                                                           J u n e  1 0 ,  2 0 1 5      That is :      xd(y) = xg(y) , for all x U ,y R                                                            (19) And  hence  d  and  g  are  (U,R)-derivations .  REFERENCES [1]  Yass,S., Strongly Derivation Pairs on Prime and Semiprime Rings, MSC. Thesis, Baghdad  University , (2010) . [2]  Herstein, I. N. , Topics in ring theory , University of Chicago Press , Chicago ,1969. [3]  Ashraf, M.,  Rehman, N. and  Quadri, M. A. , On Lie ideals  and (σ,τ) – Jordan derivations  on prime ring , Tamkang J. Math. 32(2001), 247 - 252 . [4]  Dalgin, H. , Lie ideals  and generalized derivations of  prime rings , Internat. J. of Algebra  4(10) (2010) , 461- 467 .  

( U,R) STRONGLY  DERIVATION  PAIRS  ON LIE IDEALS  IN  RINGS

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( U,R) STRONGLY DERIVATION PAIRS ON LIE IDEALS IN RINGS

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