1. 1 “DERIVATIONS AND RELATED MAPS” by Dr. M. S. Samman Dr. A. B. Thaheem

2. 2 Contents 1. Introduction 2. Definitions and notations 3. Background of the problem(s) 4. Objectives 5. Results 6. References

3. 3 1. Introduction The theory of derivations plays an important role in quantum physics. The study of bounded derivations on operator algebras started in 1950s and has been very actively carried out by many mathematicians, and it has become a useful mathematical theory. After the study of bounded derivations (see Kadison [26], Sakai [37, 38] ), the study of unbounded derivations is being carried on by many mathematicians and mathematical phycisists, and this theory is making great progress (see Bratteli and Robinson [6]). The theory of derivations plays an important role in quantum physics. The study of bounded derivations on operator algebras started in 1950s and has been very actively carried out by many mathematicians, and it has become a useful mathematical theory. After the study of bounded derivations (see Kadison [26], Sakai [37, 38] ), the study of unbounded derivations is being carried on by many mathematicians and mathematical phycisists, and this theory is making great progress (see Bratteli and Robinson [6]).

4. 4 1. Introduction (cont’d) Derivations have also been studied for Banach algebras with particular attention to the problem of automatic continuity of derivations with reference to the Singer-Wermer conjecture [39] which has bene finally resolved by Thomas [47] in 1988. Derivations have also been studied for Banach algebras with particular attention to the problem of automatic continuity of derivations with reference to the Singer-Wermer conjecture [39] which has bene finally resolved by Thomas [47] in 1988. C*-algebras are semiprime rings and factors (von Neumann algebras with center consisting of scalar multiples of the identity operator) are prime rings. Therefore, it is natural to consider derivations on prime and semiprime rings. C*-algebras are semiprime rings and factors (von Neumann algebras with center consisting of scalar multiples of the identity operator) are prime rings. Therefore, it is natural to consider derivations on prime and semiprime rings. Posner [36] initiated several aspects of a study of derivations such as composition of derivations, commuting and centralizing derivations on prime rings. Posner [36] initiated several aspects of a study of derivations such as composition of derivations, commuting and centralizing derivations on prime rings.

5. 5

6. 6 1. Introduction (cont’d) Posner [36], proved that zero is the only centralizing derivation on a non-commutative prime ring. J. Mayne [32, 33] proved the analogous result for centralizing automorphisms on prime rings. A number of authors have generalized these theorems of Posner and Mayne in several ways (see e.g. [3, 7, 10, 12, 27, 29, 48, 49]). Further, a lot of work has been done on the general theory of derivations and related generalized notions on prime and semiprime rings (see e.g. [4, 8, 9, 18, 23-25]). Posner [36], proved that zero is the only centralizing derivation on a non-commutative prime ring. J. Mayne [32, 33] proved the analogous result for centralizing automorphisms on prime rings. A number of authors have generalized these theorems of Posner and Mayne in several ways (see e.g. [3, 7, 10, 12, 27, 29, 48, 49]). Further, a lot of work has been done on the general theory of derivations and related generalized notions on prime and semiprime rings (see e.g. [4, 8, 9, 18, 23-25]). Derivations have also been extended to  -derivations and ( , β)-derivations where  and β are automorphisms. These derivations are also called skew-derivations. (see [19, 28, 44]). Derivations have also been extended to  -derivations and ( , β)-derivations where  and β are automorphisms. These derivations are also called skew-derivations. (see [19, 28, 44]).

7. 7 1. Introduction (cont’d) This work is concerned with the general theory of derivations and related maps on prime and semiprime rings. We shall summarize some results obtained in this direction. In fact, our study here deals with general properties of derivations,  -derivations, centralizing and commuting maps, a functional equation and automorphisms for prime and semiprime rings. Any extension of these and other results to near-rings remains an open problem. This work is concerned with the general theory of derivations and related maps on prime and semiprime rings. We shall summarize some results obtained in this direction. In fact, our study here deals with general properties of derivations,  -derivations, centralizing and commuting maps, a functional equation and automorphisms for prime and semiprime rings. Any extension of these and other results to near-rings remains an open problem.

8. 8 Definitions & Notations  R denotes a ring with center Z(R)  [x, y] = xy - yx for x, y in R. [xy, z] = x[y, z] + [x, z]y & [x, yz] = y[x, z] + [x, y]z for x, y, z  R.  R is prime if aRb = (0) implies a= 0 or b= 0  R is semiprime if aRa = (0) implies a = 0

9. 9 Definitions & Notations (cont’d)  An additive map d: R→ R is called a derivation if d(x y) = d(x)y + xd(y) for all x, y  R. if d(x y) = d(x)y + xd(y) for all x, y  R.  A mapping f : R→ R is called centralizing if [f(x), x]  Z(R); in particular, if [f(x), x] = 0 [f(x), x]  Z(R); in particular, if [f(x), x] = 0 for all x  R, then it is called commuting. for all x  R, then it is called commuting. It is easy to see that if f: R→R is additive and commuting map then [f(x), y] = [x, f(y)] x, y  R. Clearly, commuting → centralizing

10. 10 Definitions & Notations (cont’d)  An additive map f on a ring R is called antihomomorphism if f(xy) = f(y)f(x) for all x, y  R.  A mapping f: R→R is called commutativity- preserving if [f(x), f(y)] = 0 whenever [x, y] = 0. preserving if [f(x), f(y)] = 0 whenever [x, y] = 0.  A mapping f: R→R is called strong commutativity-preserving if [f(x), f(y)] = [x, y] for all x, y  R.  A mapping f: R→R is called strong commutativity-preserving if [f(x), f(y)] = [x, y] for all x, y  R.

11. 11 Various aspects of the derivation are in relations with operators algebras, C*-algebras, von Neumann algebras, etc. Various aspects of the derivation are in relations with operators algebras, C*-algebras, von Neumann algebras, etc. Background of the problem(s) The use of derivations in operators equations. The use of derivations in operators equations. Centralizing automorphisms have impact on the theory of C*-algebra. Centralizing automorphisms have impact on the theory of C*-algebra.

12. 12 4. Objectives VI. Derivations and related mappings in near- rings (?) I. General properties of derivations II.  -derivations on prime and semiprime rings III. Commuting and centralizing derivations IV. Commuting and centralizing automorphisms V. Certain functional equations on prime and semiprime rings

13. 13 5. Results  Results -A  Related to Results -A Related to Results -A Related to Results -A  Lemma A1: Let R be a semiprime ring, I a nonzero two-sided ideal of R and a  R such that axa = 0 for all x  I, then a = 0. Let R be a semiprime ring, I a nonzero two-sided ideal of R and a  R such that axa = 0 for all x  I, then a = 0.  Theorem A2 : Let R be a semiprime ring, I a nonzero two-sided ideal of R and f, g be derivations of R such that f(x)y + yg(x) = 0 for all x, y  I. Then f(u) [x, y] = [x,y]g(u) = 0 for all x,y,u  I; in particular, f and g map I into Z(R).

14. 14 5. Results (cont’d) - A  Corollary A3 : Let R be a noncommutative prime ring, I a nonzero two- sided ideal of R and f, g be derivations of R such that f(x) y + yg(x) = 0 for all x, y  I. Then f = g = 0 on R. f(x) y + yg(x) = 0 for all x, y  I. Then f = g = 0 on R.  Remark A4 : If R is a noncommutative prime ring, I a nonzero two-sided ideal of R such that f(x)x = 0 for all x  I, then f = 0 on R. This follows from Corollary A3. Indeed, put g = 0 and y=x in (A3), we get f(x)x = 0 for all x  I and hence f = 0 on R.  Theorem A5 : Let R be a noncommutative prime ring, I a nonzero two- sided ideal of R and f, g be derivations of R such that f(x)xy + yg(x)x = 0 for all x, y  I. Then f = g = 0. f(x)xy + yg(x)x = 0 for all x, y  I. Then f = g = 0.

15. 15 5. Results (cont’d)  Results -B  Related to Results -B Related to Results -B Related to Results -B  Theorem B1: Let T be an endomorphism of a 2-torsion free semiprime ring R such that the mapping x [T(x), x] is commuting on R. Then ([T(x), x]) 2 =0 for all x  R. commuting  Corollary B2: Let T be a centralizing endomorphism of a 2-torsion free semiprime ring R, then it is commuting on R. centralizing  Theorem B3: Let R be a 2-torsion free and 3-torsion free semiprime ring and T an endomorphism of R such that the mapping x [T(x), x] is centralizing on R. Then it is commuting on R; in particular, ([T(x), x]) 2 = 0 for all x  R. x [T(x), x] is centralizing on R. Then it is commuting on R; in particular, ([T(x), x]) 2 = 0 for all x  R.

16. 16 5. Results (cont’d)  Results -C Proposition C1: Proposition C1: Let R be a semiprime ring and f an epimorphism of R. Then f is centralizing if and only if it is strong commutativity-preserving. strong commutativity-preservingstrong commutativity-preserving Proposition C2: Proposition C2: Let R be a ring and f: R R an antihomomorphism. Then f is commutativity-preserving. Proposition C3: Proposition C3: Let R be a 2-torsion free semiprime ring and f a centralizing antihomomorphism of R onto itself. Then f is strong commutativity-preserving.

17. 17 5. Results (cont’d) - C Bresar [10, Proposition 4.1] has proved the following result. Theorem Theorem Let R be a 2-torsion free semiprime ring and f: R R be a centralizing antihomomorphism. Then i. S = {x  R: f(x) = x}  Z(R). ii. If R is prime and f does not map R into Z(R), then S = Z(R). Remark Remark We note that this Theorem can also be obtained as an application of Proposition C3 if f is onto.

18. 18 References [1]H.E. Bell and W.S. Martindale, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30(1987), 92-101. [1]H.E. Bell and W.S. Martindale, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30(1987), 92-101. [2]M. Bresar, Centralizing mappings on Von Neumann algebras, Proc. Amer. Math. Soc. III(1`991), 501-510. [2]M. Bresar, Centralizing mappings on Von Neumann algebras, Proc. Amer. Math. Soc. III(1`991), 501-510. [3] Mr. Bresar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 100(1990), 7-16. [3] Mr. Bresar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 100(1990), 7-16. [4]M. Choda, I. Kasahara and R. Nakamoto, Dependent elements of automorphisms of a C*-algebra, Proc. Japan Acad. 48(1972), 561-565. [4]M. Choda, I. Kasahara and R. Nakamoto, Dependent elements of automorphisms of a C*-algebra, Proc. Japan Acad. 48(1972), 561-565. [5]J.M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53(1975), 321-324. [5]J.M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53(1975), 321-324. [6]I.N. Herstein, Jordan derivations in prime rings, Proc. Amer. Math. Soc. 8(1957), 1104-1110. [6]I.N. Herstein, Jordan derivations in prime rings, Proc. Amer. Math. Soc. 8(1957), 1104-1110. [7]R.R. Kallman, A generalization of free action, Duke Math. J. 36(1969), 781-789. [7]R.R. Kallman, A generalization of free action, Duke Math. J. 36(1969), 781-789.

19. 19 References (cont’d) [8]L.A. Khan and A.B. Thaheem, On automorphisms of prime rings with involution, Demonst. Math. 30(1997), 307-311. [8]L.A. Khan and A.B. Thaheem, On automorphisms of prime rings with involution, Demonst. Math. 30(1997), 307-311. [9]Laradji and A.B. Thaheem, On dependent elements in semiprime rings, Math. Japonica, 47(1998), 29-31. [9]Laradji and A.B. Thaheem, On dependent elements in semiprime rings, Math. Japonica, 47(1998), 29-31. [10]J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19(1976), 113-115. [10]J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19(1976), 113-115. [11]J. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27(1984), 122-126. [11]J. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27(1984), 122-126. [12]J. D. P. Meldrum, Near-rings and their links with groups, Research Notes in Maths. 134, Pitman, London, 1985. [12]J. D. P. Meldrum, Near-rings and their links with groups, Research Notes in Maths. 134, Pitman, London, 1985. [13]E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8(1957), 1093-1100. [13]E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8(1957), 1093-1100. [14]S. Sakai, C*-algebras and W*-algebras, Springer-Verlag, Berlin, 1971. [14]S. Sakai, C*-algebras and W*-algebras, Springer-Verlag, Berlin, 1971.

20. 20 References (cont’d) [15]M.S. Samman, M.A. Chaudhry and A.B. Thaheem, A note on commutativity of automorphisms, Internat. J. Math. & Math. Sci. 21(1998), 201-204. [15]M.S. Samman, M.A. Chaudhry and A.B. Thaheem, A note on commutativity of automorphisms, Internat. J. Math. & Math. Sci. 21(1998), 201-204. [16]A.M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20(1969), 166-170. [16]A.M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20(1969), 166-170. [17]I.M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129(1955), 260-264. [17]I.M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129(1955), 260-264. [18]S. Stratila, Modular Theory of Operator Algebras, Abcus Press, Kent, 1981. [18]S. Stratila, Modular Theory of Operator Algebras, Abcus Press, Kent, 1981. [19]M. Thomas, The image of a derivation is contained in the radical, Ann. Math. 128(1988), 435-460. [19]M. Thomas, The image of a derivation is contained in the radical, Ann. Math. 128(1988), 435-460. [20]J. Vukman, Derivations on semiprime rings, Bull. Austral. Math. Soc. 53(1995), 353-359. [20]J. Vukman, Derivations on semiprime rings, Bull. Austral. Math. Soc. 53(1995), 353-359. [21]J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109(1990), 47-52. [21]J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109(1990), 47-52.

21. 21 Thanks

22. 22 Related to results -A A classical result in the theory of centralizing derivations is a theorem of Posner [36, Theorem 2] which states that noncommutative prime rings do not admit nonzero centralizing derivations. Vukman [48] has proved that if f is a derivation on a noncommutative prime ring R of characteristic not two such that the mapping x [d(x), x] is commuting on R, then d = 0. Alternatively, this result states that if d is a derivation which satisfies the identity d(x)x 2 + x 2 d(x) - 2xd(x)x = 0 for all x  R, then d = 0. This, in fact, an analog of Posner's results for derivations satisfying this identity.

23. 23 Related to results -B Certain situations have been identified where centralizing and commuting maps coincide. For instance, Bell and Martindale [3, Lemma 2] have shown that if T is an endomorphism of a semiprime ring R which is centralizing on a nonzero left ideal U of R, then T is commuting on U.