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Vudhivanich, V. and A. Rittima. 2003. Application of artificial neural networks for reservoir inflow forecasting : Lam Takong reservoir case study. Final Report. Department of Irrigation Engineering, Faculty of Engineering, Kasetsart University, Kamphaengsaen Campus, Nakhon Pathom. Kasetsart J. (Nat. Sci.) 40(1) 259 Vudhivanich, V., Thongpumnak, S., Cherdchanpipat, N., Kasempun, N. and A. Rittima. 2004. Reservoir inflow forecasting by artificial neural networks, pp. 24-31. In Proceedings of 42 nd Kasetsart University Annual Conference. Kasetsart University, Bangkok.
Kasetsart J. (Nat. Sci.) 40 : 260 - 263 (2006) Jordan Derivations on Rings ABSTRACT Orapin Wootijiruttikal and Utsanee Leerawat An additive mapping d : R→R is called a Jordan derivation on a ring R if d(a 2 ) = d(a)a + ad(a) for all a ∈ R. Two general forms of d n (aba) and d n (abc+cba), where a,b,c ∈ R and n ∈ , are established. It is also shown that if d is a Jordan derivation on a commutative ring R and P is a semiprime ideal or prime ideal of R where R/P is characteristic – free, then d(P) ⊆ P if and only if d n (P) ⊆ P for all positive integers n ≥ 2. Key words: derivation, Jordan derivation, ring INTRODUCTION An additive mapping d : R → R is called a derivation on R if d(ab) = d(a)b + ad(b) for all pairs a, b ∈ R. An additive mapping d : R → R is called a Jordan derivation if d(a 2 ) = d(a)a + ad(a) holds for all a ∈ R. Obviously, every derivation is a Jordan derivation. Herstein(1957) showed that if d is a Jordan derivation on a ring R such that characteristic is not 2, then d(aba) = d(a)ba + ad(b)a + abd(a) and d(abc+cba) = d(a )bc + ad(b)c + abd(c) + d(c)ba + cd(b)a + cbd(a) for all a,b,c ∈ R. In 1998, Creedon investigated a ring of characteristic – free, semiprime ideal on ring and proved that, if d is a derivation on a ring R and P is a semiprime ideal of R, such that R/P is characteristic – free and d k (P) ⊆ P, for any fixed positive integer k, then d(P) ⊆ P. In this paper we will consider d as a Jordan derivation on a ring R and establish general forms of d n (aba) and d n (abc+cba) for all positive integer n. We also show that if d is a Jordan derivation on a commutative ring R and P is a semiprime ideal or prime ideal of R where R/P is characteristic – free, then we have d(P) ⊆ P if and only if d n (P) ⊆ P for all positive integer n ≥ 2. MATERIALS AND METHODS For the proof results we need the following lemmas. Lemma 1. [Herstein, Lemma 3.1] Let R be a ring such that characteristic is not 2 and let d be a Jordan derivation on R. Then for all a, b, c ∈ R the following statements hold : (i) d(ab + ba) = d(a)b + ad(b) + d(b)a+ bd(a), (ii) d(aba) = d(a)ba +ad(b)a + abd(a), (iii) d(abc + cba) = d(a)bc + ad(b)c + abd(c) + d(c)ba+ cd(b)a + cbd(a). Lemma 2. [Creedon, Lemma 4 ] Suppose P is a semiprime ideal of a ring R and L is a left ideal satisfying L n ⊆ P, for some positive integer n. Then L ⊆ P. Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand. * Corresponding author, e-mail: fsciutl@ku.ac.th Received date : 17/06/05 Accepted date : 31/10/05
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January - March 2006 Volume 40 Numb
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KASETSART JOURNAL NATURAL SCIENCE T
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In sacco Degradation Characteristic
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2 and management practices a sorghu
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4 the greater plant height was obta
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6 at Wolenchity (Figure 3) that led
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8 Stover and aboveground biomass Co
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10 Radford, B.J., A.J. Dry, L.N. Ro
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12 At present, new cultivars with h
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14 2000), grapevine (Lavee, 2000),
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16 repeatability of FW, FLT, FLW, S
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18 (Table 5) indicating that select
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Kasetsart J. (Nat. Sci.) 40 : 20 -
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22 measured to define fruit shape i
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24 for higher fruit number and yiel
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28 (Table 2). The numbers of flower
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30 to the color of fruit and seed c
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32 1999. Carrots and related vegeta
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34 MATERIALS AND METHODS Laboratory
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36 DISCUSSION Extensive studies hav
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38 from Gossypium hirsutum. J. Inse
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40 The natural resistance of plants
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42 4. Data collection and statistic
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44 was not significantly different
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46 showed the highest yield compare
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48 Kessmann, H., T. Staub, C. Hofma
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50 There are approximately 60 comme
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52 in PBST containing 2% ovalbumin
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54 Kasetsart J. (Nat. Sci.) 40(1) T
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56 Virus-free orchid plants show fa
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60 feed source was based on purchas
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62 intake of DM, CP, ether extract
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64 relatively slow and short. The p
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66 week 4. Milk composition was not
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68 Agricultural Experiment Station.
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70 et al., 2002). In contrast to ma
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72 level and showed marked polymorp
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76 loam (Eutric Fluvisol) soil and
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78 Eugenia caryophyllata, Echinops
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80 CONCLUSION In conclusion, out of
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82 of chromic acid titration method
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84 initials at the time of pinning
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86 vessels a point where the number
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88 Kasetsart J. (Nat. Sci.) 40(1) T
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90 CONCLUSIONS In the investigation
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92 1994; Strand et al., 1997). Now
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94 The clones were grown overnight
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96 isozymes. The copy number for nu
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98 Helianthus annuus. Theor. Appl.
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100 MATERIALS AND METHODS Plant mat
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102 The amount of Na + (%) 6 5 4 3
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104 NGE, 7-16 folds). Finally, all
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106 Anal. Biochem. 162: 156-159. Cr
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108 grown for academic purpose at N
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110 Cluster A was further separated
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112 Table 2 Similarity index of 19
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114 subjected to UPGMA and resultin
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116 Tabel 3 The similarity index of
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118 formation as seen from phylogen
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120 Genet. 101: 860-864. Lerceteau,
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122 INTRODUCTION Methylotrophic yea
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124 RESULTS AND DISCUSSION Screenin
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126 optimum temperature, insignific
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128 medium of C. boidinii (Volfova
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130 was observed soon after 24h cul
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132 Kasetsart J. (Nat. Sci.) 40(1)
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134 ACKNOWLEDGEMENTS This work was
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138 Determination of enzyme activit
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140 Table 1 Optimum and stability o
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142 for example recombinant E. coli
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144 were studied for β-1,3-1,4-glu
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146 ACKNOWLEDGEMENT This work was s
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150 vulcanization times were calcul
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152 RESULTS AND DISCUSSION Mechanic
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154 Kasetsart J. (Nat. Sci.) 40(1)
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156 blend with 0.5 phr Irganox. How
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160 (AA-680 ShiMADZU, Atomic Absorp
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162 warm, relatively shallow and hi
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164 cycle in crustacean (Chen et al
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166 Salaenoi, J. 2004. Changes of e
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168 MATERIALS AND METHODS This rese
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170 DISCUSSION The average total am
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174 repacked and scanned three time
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176 found the most in the shrimp fe
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178 PLS model The comparison betwee
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180 brix value and dry matter of ma
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182 approach to increase immunity t
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184 higher (P
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186 levels were significantly eleva
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190 currents. Zone 4: River outlet
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192 standing waters, thus the occur
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194 and moving to 59%TL in juvenile
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198 thin membrane (Figure 1A, B). T
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200 were clearly seen. The presence
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202 could change to female at 6-12
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206 water until the water became cl
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Page 215 and 216: 210 cavalcade hay. The crude protei
Page 217 and 218: 212 CONCLUSIONS Ruminal disappeared
Page 219 and 220: 214 ∅rskov, E.R. and I. McDonald.
Page 221 and 222: 216 is constrained mainly due to fa
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Page 225 and 226: 220 in terms of both manual samplin
Page 227 and 228: 222 Harding, F. 1995. Compositional
Page 229 and 230: 224 proteins described by Barbut an
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Page 239 and 240: 234 categories including puffed sna
Page 241 and 242: 236 Cohesiveness of mass Crispness
Page 243 and 244: 238 Cohesiveness of mass Sweetness
Page 247 and 248: 242 3. Physicochemical properties 3
Page 251 and 252: 246 ACKNOWLEDGEMENTS This research
Page 253 and 254: 248 are now becoming more and more
Page 255 and 256: 250 downstream control. Lam Chae re
Page 259 and 260: 254 Inflow (mcm.) Inflow (mcm.) Inf
Page 261 and 262: 256 were above 0.70. The testing re
Page 263: 258 1999-2000 could be used for tra
Page 267 and 268: 262 we assume that r ≥ 2 and d r
Page 271 and 272: 266 Figure 1 Prototype domains. num
Page 273 and 274: 268 DISCUSSION Consequences indicat
Page 277 and 278: 272 George, B. K. 1997. The GIS Boo
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Page 281 and 282: LITERATURE CITED FORMAT Manuscripts
Page 283 and 284: PRE-SUBMISSION CHECKLIST (see “In
Page 285 and 286: Send application materials to Payme
Page 287: Text and Journal Publication Co., L
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