臺灣博碩士論文加值系統

令D是一個可除環，M 是一個係數在D上的右向量空間，End( D M )是所有 由M 到M 的線性變換形成的環，R是 End( D M )的一個稠密子環且R 中的元素 皆為有限秩的線性變換。設 R Rf  : 是一個對R 中所有秩為k 的線性變換具有 冪次交換性的加性函數，其中k 是一個滿足 D Mk dim1  的固定整數。本篇論文 的主要結果是給出函數 f 的結構定理。

Let M be a right vector space over a division ring D and let End ) ( D M be the ring of all D-linear transformations from M into M . Suppose that R is a dense subring of End ) ( D M consisting of finite rank transformations and RRf : is an additive map satisfying ) ()( )()( x fxxxf xmxm  for every rank-k transformation R x , where k is a fixed integer with D Mk dim1  and 1)( xm is an integer depending on x. Then there exist ) (DZ  and an additive map I DZR ) (:   such that ) ()( x xxf   for all R x , where I denotes the identity transformation on M .

1. Introduction and Results ................. 1
2. Proofs ................................... 4
References .................................. 23

[1] K.I. Beidar, On functional identities and commuting additive mappings, Comm. Alge- bra 26 (1998), 1819–1850.
[2] K.I. Beidar, Y. Fong, P.-H. Lee and T.-L. Wong, On additive maps of prime rings satisfying the Engel condition, Comm. Algebra 25 (1997), 3889–3902.
[3] K. I. Beidar, W. S. Martindale 3rd and A. V. Mikhalev, “Rings with Generalized Identities”, Marcel Dekker, Inc., New York–Basel–Hong Kong, 1996.
[4] M. Breˇsar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385–394.
[5] M. Breˇsar, On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47 (1993), 291–296.
[6] M. Breˇsar and Hvala, On additive maps of prime rings, Bull. Austral. Math. Soc. 51 (1995), 377–381.
[7] M. Breˇsar, M. A. Chebotar and W. S. Martindale III (2007), “Functional Identities”, Frontiers in Mathematics. Basel: Birkhauser Verlag. [8] M. Breˇsar and ˇ S. ˇSpenko, Functional identities in one variable, J. Algebra 401 (2014), 234–244.
[9] M. Chacron, Involution satisfying a local Engel or power commuting condition, Comm. Algebra 45 (2017), 2018–2028.
[10] C.-W. Chen, M.-T. Ko¸san, T.-K. Lee, Decompositions of quotient rings and m-power commuting maps, Comm. Algebra 41 (2011), 1865–1871.
[11] B. Dhara and S. Ali, On n-centralizing generalized derivations in semiprime rings with applications to C∗-algebras, J. Algebra Appl. 11 (2012), 1250111 [11 pages].
[12] W. Franca, Commuting maps on some subsets of matrices that are not closed under addition, Linear Algebra Appl. 437 (2012), 388–391.
[13] W. Franca, Commuting maps on rank-k matrices, Linear Algebra Appl. 438 (2013), 2813–2815.
[14] W. Franca, Commuting traces of multiadditive maps on invertible and singular matri- ces, Linear Multilinear Algebra 61 (2013), 1528–1535.
[15] W. Franca, Commuting traces on invertible and singular operators, Oper. Matrices 9 (2015), 305–310.
[16] W. Franca, Commuting traces of biadditive maps on invertible elements, Comm. Al- gebra 44 (2016), 2621–2634.
[17] W. Franca, Weakly commuting maps on the set of rank-1 matrices, Linear Multilinear Algebra 65 (2017), 479–495.
[18] W. Franca and N. Louza, Commuting maps on rank-1 matrices over noncommutative division rings, Comm. Algebra 45 (2017), 4696–4706.
[19] H.G. Inceboz, M.-T. Ko¸san and T.-K. Lee, m-Power commuting maps on semi-prime rings, Comm. Algebra 42 (2014), 1095–1110.
[20] T.-K. Lee, K.-S. Liu and W.-K. Shiue, n-Commuting maps on prime rings, Publ. Math. Debrecen 63 (2004), 463–857.
[21] C.-K. Li and S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), 591–605.
[22] C.-K. Liu, Strong commutativity preserving maps on some subsets of matrices that are not closed under addition, Linear Algebra Appl. 458 (2014), 280–290.
[23] C.-K. Liu, Centralizing maps on invertible or singular matrices over division rings, Linear Algebra Appl. 440 (2014), 318–324.
[24] C.-K. Liu and J.-J. Yang, Power commuting additive maps on invertible or singular matrices, Linear Algebra Appl. 530 (2017), 127–149.
[25] X. Xu, L. Chen and J. Zhu, Maps determined by rank-s matrices for relatively small s, Aequationes Math. 91 (2017), 391–400.
[26] X. Xu and H. Liu, Additive maps on rank-s matrices, Linear Multilinear Algebra 65 (2017), 806–812.
[27] X. Xu and X. Yi, Commuting maps on rank-k matrices, Electron. J. Linear Algebra 27 (2014), 735–741.
[28] X. Xu, Y. Pei and X. Yi, Additive maps on invertible matrices, Linear Multilinear Algebra 64 (2016), 1283–1294.