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1. Generalized @U; M A-Derivations in Prime -Rings
M. M. Rahman
1
& A. C. Paul
2
1
Corresponding Author
Department of Mathematics, Jagannath University
Dhaka, Bangladesh; e-mail: mizanorrahman@gmail.com
2
Department of Mathematics, University of Rajshahi
Rajshahi, Bangladesh; e-mail: acpaulrubd math@yahoo.com
Abstract
vet M ˜e — PEtorsion free prime Ering s—tisfying the ™ondition ab

2. c a a

3. bc;8a;b;c 2 M
—nd ;

4. 2 D U ˜e — vie ide—l of M —nd f ˜e — gener—lized @U;MAEderiv—tion of MF „hen we
prove the following resultsX
IF sf U is —n —dmissi˜le vie ide—l of MD then f@uvA a f@uAv C ud@vA;8u;v 2 U; 2 F
PF sf uu 2 U;8u 2 U; 2 D then f@umA a f@uAm C ud@mA;8u 2 U;m 2 M; 2 D
where d is — @U;MAEderiv—tion of MF
KeywordsX heriv—tionD vie ide—lD —dmissi˜le vie ide—lD gener—lized @U;MAEderiv—tionD EringD prime
EringF
2010 AMS Subject Classi

5. cationX €rim—ry IQxISY ƒe™ond—ry IT‡IHD IUgSHF
1 Introduction
sn ‘W“D rerstein proved — wellEknown result in prime rings th—t every tord—n deriv—tion is — deriv—E
tionF efterw—rds m—ny w—them—ti™i—ns studied extensively the deriv—tions in prime ringsF sn ‘Q“D
ewt—r extended this result in vie ide—lsF @U;RAEderiv—tions in rings h—ve ˜een introdu™ed ˜y p—r—jD
r—etinger —nd w—jeed ‘U“D —s — gener—liz—tion of tord—n deriv—tions on — vie ide—ls of — ringF „he
notion of — @U;RAEderiv—tion extends the ™on™ept given in ‘Q“F sn this p—per ‘U“D they proved th—t if
R is — prime ringD ™h—r@RA Ta PD U — squ—re ™losed vie ide—l of R —nd d — @U;RAE deriv—tion of RD
then d@urA a d@uAr C ud@rA;V;u P U;r P RF „his result is — gener—liz—tion of — result in ewt—r ‘QD
„heorem in se™tion Q“F
„he notion of — Ering h—s ˜een developed ˜y xo˜us—w— ‘IQ“D —s — gener—liz—tion of — ringF pollowE
ing f—rnes ‘R“ gener—lized the ™on™ept of xo˜us—w—9s Ering —s — more gener—l n—tureF xow — d—ysD
Ering theory is — showpie™e of m—them—ti™—l uni™—tionD ˜ringing together sever—l ˜r—n™hes of the
su˜je™tF st is the ˜est rese—r™h —re— for the w—them—ti™i—ns —nd during RH ye—rsD m—ny ™l—ssi™—l ring
theories h—ve ˜een gener—lized in Erings ˜y m—ny —uthorsF
WV

6. „he notions of deriv—tion —nd tord—n deriv—tion in Erings h—ve ˜een introdu™ed ˜y ƒ—p—n™i —nd
x—k—jim— ‘IR“F efterw—rdsD in the light of some signi™—nt results due to tord—n left deriv—tion of —
™l—ssi™—l ring o˜t—ined ˜y tun —nd uim in ‘II“D some extensive results of left deriv—tion —nd tord—n
left deriv—tion of — Ering were determined ˜y geven in ‘T“F sn ‘V“D r—lder —nd €—ul extended the
results of ‘T“ in vie ide—lsF
sn this —rti™leD we introdu™e the ™on™ept of @U;MAEderiv—tion —nd gener—lized @U;MAEderiv—tionD
where U is — vie ide—l of — Ering M F en ex—mple of — vie ide—l of — Ering —nd —n ex—mple of
@U;MAEderiv—tion —nd gener—lized @U;MAEderiv—tion —re given hereF e result in ‘UD „heorem PFV“ is
gener—lized in Erings ˜y the new ™on™ept of — @U;MAEderiv—tionF
vet M —nd ˜e —dditive —˜eli—n groupsF sf there is — m—pping
M ¢ ¢M 3 M @sending @x;;yA into xyA su™h th—t
@iA @x C yAz a xz C yz;
x@ C

7. Ay a xy C x

8. y;
x@y C zA a xy C xzD
@iiA @xyA

9. z a x@y

10. zA;
for —ll x;y;z P M —nd ;

11. P D then M is ™—lled — -ringF „his ™on™ept is more gener—l th—n — ring
—nd w—s introdu™ed ˜y f—rnes ‘R“F e Ering M is ™—lled — prime -ring if Va;b P M;a M b a H
implies a a H or b a H —nd M is ™—lled semiprime if a M a a H @with a P MA implies a a HF e
Ering M is 2-torsion free if Pa a H implies a a H;Va P M:
por —ny x;y P M —nd P D we indu™e — new produ™t D the Lie product ˜y ‘x;y“ a xy yxF
en —dditive su˜group U M is s—id to ˜e — Lie ideal of M if whenever u P U;m P M —nd P D
then ‘u;m“ P UF
sn the m—in results of this —rti™le we —ssume th—t the vie ide—l … veries uu P U;Vu P UF e vie
ide—l of this type is ™—lled — square closed Lie idealF
purthermoreD if the vie ide—l U is squ—re ™losed —nd U is not ™ont—ined in Z@MADwhere Z@MAdenotes
the ™enter of MDthen U is ™—lled —n admissible Lie ideal of MF
vet M ˜e — EringF en —dditive m—pping d X M 3 M is ™—lled — derivation if d@abA a d@aAb C
ad@bA;Va;b P M —nd P F
en —dditive m—pping d X M 3 M is ™—lled — Jordan derivation if
d@aaA a d@aAa C ad@aA;Va P M —nd P F
„hroughout the —rti™leD we use the ™ondition ab

12. c a a

13. bc;Va;b;c P M —nd ;

14. P —nd this is
represented ˜y @BAF
‡e m—ke the ˜—si™ ™ommut—tor identitiesX
‘xy;z“

15. a ‘x;z“

16. y C x‘;

17. “zy C x‘y;z“

18. F
—nd ‘x;yz“

19. a ‘x;y“

20. z C y‘;

21. “xz C y‘x;z“

22. D Va;b;c P M —nd V;

23. P F
e™™ording to the ™ondition @BAD the —˜ove two identities redu™es to
‘xy;z“

24. a ‘x;z“

25. y C x‘y;z“

26. F
—nd ‘x;yz“

27. a ‘x;y“

28. z C y‘x;z“

29. ;Va;b;c P M —nd V;

30. P F
WV

31. 2 Generalized (U; M )-Derivation
2.1 De

32. nition: vet M ˜e — Ering —nd U ˜e — vie ide—l of MF en —dditive m—pping d X M 3 M is
s—id to ˜e — @U;MA- derivation of M if d@umCsuA a d@uAmCud@mACd@sAuCsd@uA;Vu P
U;m;s P M —nd P F
2.2 De

33. nition: vet M ˜e — Ering —nd U ˜e — vie ide—l of MF en —dditive m—pping f X M 3 M
is s—id to ˜e — generalized @U;MA- derivation of M if there exists — @U;MAEderiv—tion d of M su™h
th—t f@um C suA a f@uAm C ud@mA C f@sAu C sd@uA;Vu P U;m;s P M —nd P F
„he existen™e of — vie ide—l of — EringD @U;MAEderiv—tion —nd — gener—lized @U;MAEderiv—tion —re
™onrmed ˜y the following ex—mplesX
2.3 Example: vet R ˜e — ™ommut—tive ring with ™h—r—™teristi™ P h—ving unity element IF vet
M a MP;P@RA —nd
a
nI:I nQ:I
nP:I nR:I
X ni P @Z PZA;i a I;P;Q;RYnI a nR;nP a nQ
F
„hen w is — EringF
vet U a
x y
y x
X x;y P R
F
„hen U is — vie ide—l of MF
vet us dene — m—pping f X M 3 M ˜y
f
a b
c d
a
a H
H d
;V
a b
c d
P M
„hen there exists — @U;MAEderiv—tionD d of M whi™h is dened ˜y
d
a b
c d
a
H b
c H
;V
a b
c d
P M
„hen f is — gener—lized @U;MAEderiv—tion of MF
2.4 Lemma: vet M ˜e — PEtorsion free Ering s—tisfying the ™ondition @BAF U ˜e — vie ide—l of
M —nd f ˜e — gener—lized @U;MAEderiv—tion of MF„hen
@iA f@um

34. uA a f@uAm

35. u C ud@mA

36. u C um

37. d@uA;Vu P U;m P M —nd ;

38. P F
@iiA f@um

39. vCvm

40. uA a f@uAm

41. vCud@mA

42. vCum

43. d@vACf@vAm

44. uCvd@mA

45. uCvm

46. d@uA;Vu;v P
U;m P M —nd ;

47. P F
Proof: fy the denition of gener—lized @U;MAEderiv—tion of MDwe h—ve
f@um C suA a f@uAm C ud@mA C f@sAu C sd@uA;Vu P U;m;s P M —nd P F
‚epl—™ing m —nd s ˜y @PuA

48. m C m

49. @PuA —nd let
w a u@@PuA

50. m C m

51. @PuAA C @@PuA

52. m C m

53. @PuAAuF
„hen ˜y using @BA
f@wA a P@f@uA@u

54. m C m

55. uA C ud@u

56. m C m

57. uA C f@u

58. m C m

59. uAu C @u

60. m C m

61. uAd@uAA
a P@f@uAu

62. m C f@uAm

63. u C ud@uA

64. m C uu

65. d@mA C ud@mA

66. u C um

67. d@uA C f@uA

68. mu C
u

69. d@mAu C f@mA

70. uu C m

71. d@uAu C u

72. md@uA C m

73. ud@uAA
a P@f@uAu

74. m C f@uAm

75. u C ud@uA

76. m C uu

77. d@mA C ud@mA

78. u C um

79. d@uA C f@uAm

80. u C
ud@mA

81. u C f@mAu

82. u C md@uA

83. u C um

84. d@uA C mu

85. d@uAA::::::@IAF
yn the other h—nd
WW

86. f@wA a f@@PuuA

87. m C m

88. @PuuAA C Pf@u

89. muA C Pf@um

90. uA
a P@f@uAu

91. m C ud@uA

92. m C uu

93. d@mA C f@mA

94. uu C m

95. d@uAu C m

96. ud@uA C Rf@um

97. uA
a P@f@uAu

98. mCud@uA

99. mCuu

100. d@mACmd@uA

101. uCmu

102. d@uACf@mAu

103. uACRf@um

104. uA::::::@PA
fy ™omp—ring @IA —nd @PA —nd sin™e M is PEtorsion freeD we o˜t—in
f@um

105. uA a f@uAm

106. u C ud@mA

107. u C um

108. d@uA::::::@QA;
Vu P U;m P M —nd ;

109. P F
sf we line—rize @QA on uD then @iiA is o˜t—inedF
2.5 De

110. nition: vet d ˜e — @U;MAEderiv—tion of MD then we dene ¨@u;mA a d@umA d@uAm
ud@mA
Vu P U;m P M —nd P F
2.6 Lemma: vet d ˜e — @U;MAEderiv—tion of MD then
@iA¨@u;mA a ¨@m;uAD Vu P U;m P M —nd P F
@iiA ¨@u C v;mA a ¨@u;mA C ¨@v;mA;Vu;v P U;m P M —nd P F
@iiiA ¨@u;m C nA a ¨@u;mA C ¨@u;nA;Vu P U;m;n P M —nd P F
@ivA ¨C

111. @u;mA a ¨@u;mA C ¨

112. @u;mA;Vu P U;m P M —nd ;

113. P F
„he proofs —re o˜vious ˜y using the denition PFS
2.7 De

114. nition: sf f is — gener—lized @U;MAEderiv—tion of M —nd d is — @U;MAEderiv—tion of MD
then we dene ©@u;mA a f@umA f@uAm ud@mA;Vu P U;m P M —nd P F
2.8 Lemma: vet f ˜e — gener—lized @U;MAEderiv—tion of MD then
@iA©@u;mA a ©@m;uAD Vu P U;m P M —nd P F
@iiA ©@u C v;mA a ©@u;mA C ©@v;mA;Vu;v P U;m P M —nd P F @iiiA ©@u;m C nA a
©@u;mA C ©@u;nA;Vu P U;m;n P M —nd P F
@ivA ©C

115. @u;mA a ©@u;mA C ©

116. @u;mA;Vu P U;m P M —nd ;

117. P F
„he proofs —re o˜vious ˜y using the denition PFU
2.9 Lemma: vet M ˜e — PEtorsion free prime Ering s—tisfying the ™ondition @BAD U —n —dmissi˜le
vie ide—l of M —nd f — gener—lized @U;MAE deriv—tion of M then ©@u;vA

118. w
‘u;v“ a H;Vu;v;w P U
—nd ;

119. ;
P F
Proof: vet x a R@uv

120. w
vu C vu

121. w
uvAF
„hen ˜y using vemm— PFR@iiAD we h—ve
f@xA a f@@PuvA

122. w
@PvuA C @PvuA

123. w
@PuvAA
a f@PuvA

124. w
PvuCPuv

125. d@wA
PvuCPuv

126. w
d@PvuACf@PvuA

127. w
PuvCPvu

128. d@wA
PuvC
Pvu

129. w
d@PuvA
yn the other h—ndD ˜y using vemm— PFR@iADwe h—ve
f@xA a f@u@Rv

130. w
vAu C v@Ru

131. w
uAvA
a f@uARv

132. w
vu C ud@Rv

133. w
vAu C uRv

134. w
vd@uA C
f@vARu

135. w
uv C vd@Ru

136. w
uAv C vRu

137. w
ud@vA
a Rf@uAv

138. w
vu C Rud@vA

139. w
vu C Ruv

140. d@wA
vu C
Ruv

141. w
d@vAu C Ruv

142. w
vd@uA C Rf@vAu

143. w
uv C
Rvd@uA

144. w
uv C Rvu

145. d@wA
uv C Rvu

146. w
d@uAv C
Rvu

147. w
ud@vAF
IHH

148. gomp—ring the right side of f@xA —nd using the PEtorsion freeness of M
f@uvA

149. w
vuCuv

150. w
d@vuACf@vuA

151. w
uvCvu

152. w
d@uvA a f@uAv

153. w
vuCud@vA

154. w
vuC
uv

155. w
d@vAuCuv

156. w
vd@uACf@vAu

157. w
uvCvd@uA

158. w
uvCvu

159. w
d@uAvCvu

160. w
ud@vAF
„herefore
@f@uvA f@uAv ud@vAA

161. w
vu C @f@vuA f@vAu vd@uAA

162. w
uv C uv

163. w
@d@vuA
d@vAu vd@uAA C vu

164. w
@d@uvA d@uAv ud@vAA a H
fy using the denitions PFS —nd PFUD we o˜t—in
©@u;vA

165. w
vu C ©@v;uA

166. w
uv
C uv

167. w
¨@v;uA C vu

168. w
¨@u;vA a H
xow using vemm— PFT@iA—nd PFV@iAD we h—ve
©@u;vA

169. w
‘u;v“ C ‘u;v“

170. w
¨@u;vA a H;Vu;v;w P U;;

171. ;
P
ƒin™e d is — @U;MAEderiv—tionD we h—ve ¨@u;vA a H;Vu;v P U —nd P F …sing this we o˜t—in the
desired resultF
2.10 Lemma: vet U ˜e — vie ide—l of — PEtorsion free prime Ering M —nd U is not ™ont—ined
in Z@MAF „hen there exists —n ide—l I of M su™h th—t ‘I;M“ U ˜ut ‘I;M“ is not ™ont—ined in
Z@MAF
Proof: ƒin™e M is PEtorsion free —nd U is not ™ont—ined in Z@MAD it follows from the result in ‘I“
th—t ‘U;U“ Ta H —nd ‘I;M“ UDwhere I a I ‘U;U“ M Ta H is —n ide—l of M gener—ted ˜y ‘U;U“ F
xow U is not ™ont—ined in Z@MA implies th—t ‘I;M“ is not ™ont—ined in Z@MAY for if ‘I;M“
Z@MAD then ‘I;‘I;M“ “ a HD whi™h implies th—t I Z@MA —nd hen™e I Ta H is —n ide—l of MD so
M a Z@MAF
2.11 Lemma: vet U ˜e — vie ide—l of — PEtorsion free prime Ering M s—tisfying the ™ondition @BA —nd
U is not ™ont—ined in Z@MAF sf a;b P M @respFb P U —nd a P MA su™h th—t aU

172. b a H;V;

173. P D
then a a H or b a HF
Proof: fy vemm— PFIHD there exists —n ide—l I of M su™h th—t ‘I;M“ U —nd ‘I;M“ is not ™onE
t—ined in Z@MAF xow t—ke u P U;c P I;m P M —nd ;

174. ;
P D we h—ve ‘ca

175. u;m“ P ‘I;M“ U
—nd so
H a a‘ca

176. u;m“
b;V; P :
a a‘ca;m“

177. ub C aca

178. ‘u;m“
bD ˜y using @BA
a a‘ca;m“

179. ub sin™e a

180. ‘u;m“
b P a

181. Ub a H
a a@ca
m m
caA

182. ub
a aca
m

183. ub am
ca

184. ub
a aca
m

185. ubD ˜y using —ssumption a

186. ub a H
„hus aIa
M

187. Ub a HF sf a Ta HD then ˜y the primeness of M;Ub a HF
xow if u P U —nd m P MD then ‘u;m“ P U;V P F
ren™e ‘u;m“

188. b a H;V

189. P F ƒin™e mu

190. b a H;um

191. b a HF
ƒin™e U Ta HD we must h—ve b a HF
sn the simil—r m—nnerD it ™—n ˜e shown th—t if b Ta HD then a a HF
IHI

192. 2.12 Lemma: vet M ˜e — PEtorsion free prime Ering —nd U —n —dmissi˜le vie ide—l of MF vet
GI;GP;:::;Gn ˜e —dditive groupsD S X GI ¢ GP ¢ ::: ¢ Gn 3 M —nd T X GI ¢ GP ¢ ::: ¢ Gn 3 M
˜e m—ppings whi™h —re —dditive in e—™h —rgumentF sf S@aI;:::;anA

193. x
T@aI;:::;anA a HD for every
x P U;ai P G;i a I;P;:::;n;;

194. ;
P D then S@aI;:::;anA

195. x
T@bI;:::;bnA a H
Proof: st su™es to prove the ™—se n a IF
„he gener—l proof is o˜t—ined ˜y indu™tion on nF
sf S@aA

196. x
T@aA a HD for every u P U;a P GID we get
@T@aA

197. x
S@aAAy@T@aA

198. x
S@aAA a HD for —ll x;y P U —nd ; P F
„hen ˜y vemm— PFIID T@aA

199. x
S@aA a HD for every x P U;a P GI —nd

200. ;
P F
xow line—rizing T@aA

201. x
S@aA a H we o˜t—in
S@aA

202. x
T@bA C S@bA

203. x
T@aA a HD for every x P U;a;b P GI:
ren™e @S@aA

204. x
T@bAAy@@S@aA

205. x
T@bAA
a S@aA

206. x
T@bAyS@bA

207. x
T@aA a H;Vx;y P UF
fy vemm— PFIID S@aA

208. x
T@bA a H
ƒimil—rly we ™—n prove th—t T@bA

209. x
S@aA a H;Va;b P GI —nd ;

210. ;
P F
€utting C for in the equ—tion S@aA

211. x
T@bA a H —nd using vemm— PFT@ivAD we h—ve
S@aA

212. x
T@bA C S@aA

213. x
T@bA a HF
„hereforeD we h—ve @S@aA

214. x
T@bAAy@S@aA

215. x
T@bAA
a S@aA

216. x
T@bAy@S@aA

217. x
T@bA a H
ren™e ˜y vemm— PFIID S@aA

218. x
T@bA a HF
2.13 Theorem: vet M ˜e — PEtorsion free prime Ering s—tisfying the ™ondition @BAD U ˜e —n —dE
missi˜le vie ide—l of M —nd f ˜e — gener—lized @U;MAE deriv—tion of MD then ©@u;vA a H;Vu;v P U
—nd P F
Proof: fy vemm— PFWD we h—ve ©@u;vA

219. w
‘u;v“ a H;Vu;v;w P U —nd ;

220. ;
P F
…sing the vemm— PFIP in the —˜ove rel—tionD we o˜t—in
©@u;vA

221. w
‘x;y“ a H;Vu;v;w;x;y P U —nd ;

222. ;
; P F
ƒin™e U is not ™ont—ined in Z@MAD ‘x;y“ Ta H
ƒo ˜y vemm— PFIID we get ©@u;vA a H;Vu;v P U —nd P F
Remark: sf we repl—™e U ˜y — squ—re ™losed vie ide—l in the theorem PFIQD then the theorem
is —lso trueF
xow we —re in position to prove the m—in resultF
2.14 Theorem: vet M ˜e — PEtorsion free prime Ering s—tisfying the ™ondition @BAD U — squ—re
™losed vie ide—l of M —nd f ˜e — gener—lized @U;MAE deriv—tion of MD then f@umA a f@uAm C
ud@mA;Vu P U m P M —nd P F
Proof: prom „heorem PFIQ —nd the ‚em—rk —fter the „heorem PFIQD we h—ve ©@u;vA a H;Vu;v P U
—nd P FFFFFF@eA
‚epl—™e v ˜y u

223. m m

224. u in @eAD we get
©@u;u

225. m m

226. uA a HF
ƒin™e u

227. m m

228. u P U;Vu P UD m P M —nd ;

229. P F
„herefore H a ©@u;u

230. m m

231. uA
a f@u@u

232. m m

233. uAA f@uA@u

234. m m

235. uA ud@u

236. m m

237. uA
a f@uu

238. mA f@um

239. uA f@uAu

240. m C f@uAm

241. u ud@uA

242. m uu

243. d@mA C ud@mA

244. u C
IHP

245. um

246. d@mA
a f@uu

247. mA f@uAm

248. u ud@mA

249. u um

250. d@uA f@uAu

251. m C f@uAm

252. u ud@uA

253. m
uu

254. d@mA C ud@mA

255. u C um

256. d@mA
a f@uu

257. mA f@uAu

258. m ud@uA

259. m uu

260. d@mA
„his implies th—t
f@uu

261. mA a f@uAu

262. m ud@uA

263. m uu

264. d@mAFFFFFF@fA
xow let x a uu

265. m C u

266. muF
„hen ˜y the denition of gener—lized @U;MAEderiv—tionD we h—ve
f@xA a f@uAu

267. m C ud@u

268. mA C f@u

269. mAu C u

270. md@uA
a f@uAu

271. m C ud@uA

272. m C uu

273. d@mA C f@u

274. mAu C u

275. md@uAFFFFFF@gA
yn the other h—nd
f@xA a f@uu

276. mA C f@u

277. muA
a f@uAu

278. m C ud@uA

279. m C uu

280. d@mA C f@uA

281. mu C u

282. d@mAu C u

283. md@uAFFFFFF@hA
gomp—ring @gA —nd @hA
@f@u

284. mA f@uA

285. m u

286. d@mAAu a H
„his gives
©

287. @u;mAu a H;Vu P U;m P M —nd ;

288. P FFFFFF@iA
vine—rize @iA on u —nd using equ—tion @iAD we get
©

289. @u;mAv C ©

290. @v;mAu a HFFFFFF@pA
‚epl—™e v ˜y v
v in equ—tion @pAD we o˜t—in
©

291. @u;mAv
v C ©

292. @v
v;mAu a HF
ƒin™e ©

293. @v
v;mA a H;Vv P U;m P M —nd

294. ;
P
„his is seen in the equ—tion @fA for v
v in pl—™e of uuF
„hereforeD we h—ve
©

295. @u;mAv
v a H;Vu;v P U;m P M —nd ;

296. ;
P FFFFFF@qA
sf U is non™entr—lD then repl—™e v ˜y u C v in equ—tion @qA to o˜t—in
©

297. @u;mA@u C vA
@u C vA a H
„his implies th—t ©

298. @u;mAv
u a H;Vu;v P U;m P M —nd ;

299. ;
P :
fy vemm— PFIID ©

300. @u;mA a HD sin™e U is non™entr—lF
References
‘I“ wust—f— es™i —nd ƒ—hin ger—nD „he ™ommut—tivity in prime g—mm— rings with left deriv—E
tionsD International Mathematical Forum, 2(3)(2007), 103-108.
‘P“ woh—mm—d eshr—f —nd x—deemE…rE‚ehm—nD On Lie ideals and Jordan left derivations of
prime ringsD er™hFw—thF@frnoADQT@PHHHADPHIEPHTF
‘Q“ ‚F ewterD Lie ideals and Jordan derivations of prime ringsD emerFw—thFƒo™F WH@IAD@IWVRADWE
IRF
‘R“ ‡FiF f—rnesD On the -rings of NobusawaD €—™i™ tF w—thFD IV@IWTTAD RIIERPPF
‘S“ wF fres—r —nd tF†ukm—nD Jordan derivations on prime ringsD fullFeustr—lFw—thFƒo™FD
QR@IWVVADQPIEQPPF
IHQ

301. ‘T“ ‰F gevenD Jordan left derivations on completely prime gamma ringsD gF…F penEide˜iy—t
p—kultesi pen filimleri hergisi PQ@PHHPADQWERQF
‘U“ eFuFp—r—jD gFr—etinger —nd eFrFw—jeedD Generalized Higher (U,R)-DerivationsD t€ tourn—l
of elge˜r—D IT@PAD @PHIHAD IIWEIRPF
‘V“ eFuFr—lder —nd eFgF€—ulD Jordan left derivations on Lie ideals of prime -ringsD €unj—˜
…niversity tFw—thFD@PHIIADHIEHUF
‘W“ sFxFrersteinD Jordan derivations of prime ringsD €ro™F emerF w—thF ƒo™FD V@IWSUAD IIHREIIIHF
‘IH“ sFxFrersteinD Topics in Ring TheoryD „he …niversity of ghi™—go €ressDghi™—goDIIIFE
vondonDIWTWF
‘II“ uF‡F tun —nd fFhFuimD A note on Jordan left derivationsD fullFuore—n
w—thFƒo™FDQQ@IWWTADPPIEPPVF
‘IP“ rFu—nd—m—rD The k-derivations of a gamma ringsD „urkish tF w—thFD PR @PHHHADPPIEPQIF
‘IQ“ xF xo˜us—w—D On the generalizeation of the ring theoryD ys—k— tF w—thF I@IWTRAD VIEVWF
‘IR“ wF ƒ—p—n™i —nd eF x—k—jim—D Jordan Derivations on Completely Prime -RingsD w—thF
t—poni™— RTD @IWWUAD RUESIF
VIHR