Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-16T07:17:32.667Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive n-commuting maps on semiprime rings

Part of: Radicals and radical properties of rings Rings with polynomial identity

Published online by Cambridge University Press:  11 November 2019

Cheng-Kai Liu
Cheng-Kai Liu*
Affiliation:
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan, Republic of China (ckliu@cc.ncue.edu.tw)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

Keywords

commuting mappolynomial identityfunctional identity(semi)prime ringC*-algebraorthogonal completion

MSC classification

Primary: 16N60: Prime and semiprime rings
Secondary: 16R20: Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16R60: Functional identities
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 193 - 216
Copyright
Copyright © Edinburgh Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ara, P. and Mathieu, M., An application of local multipliers to centralizing mappings of C*-algebras, Q. J. Math. 44 (1993), 129–138.CrossRefGoogle Scholar
2.Basudeb, D. and Shakir, A., On n-centralizing generalized derivations in semiprime rings with applications to C*-algebras, J. Algebra Appl. 11 (2012), 1250111.Google Scholar
3.Beidar, K. I., On functional identities and commuting additive mappings, Commun. Algebra 26 (1998), 1819–1850.10.1080/00927879808826241CrossRefGoogle Scholar
4.Beidar, K. I. and Mikhalev, A. V., Orthogonal compeleteness and algebraic systems, Russian Math. Surveys 40 (1985), 51–95.10.1070/RM1985v040n06ABEH003702CrossRefGoogle Scholar
5.Beidar, K. I. and Martindle III, W. S., On functional identities in prime rings with involution, J. Algebra 203 (1998), 491–532.10.1006/jabr.1997.7285CrossRefGoogle Scholar
6.Beidar, K. I., Martindale III, W. S. and Mikhalev, A. V., Rings with generalized identities (Marcel Dekker Inc., New York–Basel–Hong Kong, 1996).Google Scholar
7.Beidar, K. I., Fong, Y, Lee, P.-H. and Wong, T.-L., On additive maps of prime rings satisfying the Engel condition, Commun. Algebra 25 (1997), 3889–3902.CrossRefGoogle Scholar
8.Brešar, M., Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc. 111 (1991), 501–510.10.1090/S0002-9939-1991-1028283-2CrossRefGoogle Scholar
9.Brešar, M., On a generalization of the notion of centralizing mappings, Proc. Amer. Math. Soc. 114 (1992), 641–649.CrossRefGoogle Scholar
10.Brešar, M., Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385–394.CrossRefGoogle Scholar
11.Brešar, M., On certain pairs of functions of semiprime rings, Proc. Amer. Math. Soc. 120 (1994), 709–713.CrossRefGoogle Scholar
12.Brešar, M., Applying the theorem on functional identities, Nova J. Math. Game Theory Algebra 4 (1996), 43–54.Google Scholar
13.Brešar, M., Commuting maps: a survey, Taiwanese J. Math. 8 (2004), 361–397.CrossRefGoogle Scholar
14.Brešar, M. and Miers, C. R., Commuting maps on Lie ideals, Commun. Algebra 23 (1995), 5539–5553.10.1080/00927879508825551CrossRefGoogle Scholar
15.Brešar, M. and Špenko, Š., Functional identities in one variable, J. Algebra 401 (2014), 234–244.10.1016/j.jalgebra.2013.11.026CrossRefGoogle Scholar
16.Brešar, M., Martindle III, W. S. and Miers, C. R., Centralizing maps in prime rings with involution, J. Algebra 161 (1993), 342–357.10.1006/jabr.1993.1223CrossRefGoogle Scholar
17.Brešar, M., Chebotar, M. A. and Martindale III, W. S., Functional identities, Frontiers in Mathematics (Birkhauser Verlag, Basel, 2007).10.1007/978-3-7643-7796-0CrossRefGoogle Scholar
18.Chacron, M., Commuting involution, Commun. Algebra 44 (2016), 3951–3965.10.1080/00927872.2015.1087546CrossRefGoogle Scholar
19.Chacron, M., Involution satisfying an Engel condition, Commun. Algebra 44 (2016), 5058–5073.10.1080/00927872.2015.1130145CrossRefGoogle Scholar
20.Chacron, M., Involution satisfying a local Engel or power commuting condition, Commun. Algebra 45 (2017), 2018–2028.10.1080/00927872.2016.1226879CrossRefGoogle Scholar
21.Chacron, M., More on involutions with local Engel or power commuting conditions, Commun. Algebra 45 (2017), 3503–3514.10.1080/00927872.2016.1237641CrossRefGoogle Scholar
22.Chebotar, M. A., A note on certain subrings and ideals of prime rings, Commun. Algebra 26 (1998), 107–116.CrossRefGoogle Scholar
23.Chen, C.-W., Koşan, M.-T. and Lee, T.-K., Decompositions of quotient rings and m-power commuting maps, Commun. Algebra 41 (2011), 1865–1871.10.1080/00927872.2011.651764CrossRefGoogle Scholar
24.Cheung, W. S., Commuting maps of triangular algebras, J. Lond. Math. Soc. 63 (2001), 117–127.10.1112/S0024610700001642CrossRefGoogle Scholar
25.Chou, M.-C. and Liu, C.-K., An Engel condition with skew derivations, Monatsh. Math. 158 (2009), 259–270.CrossRefGoogle Scholar
26.Chuang, C.-L., The additive subgroup generated by a polynomial, Israel J. Math. 59 (1987), 98–106.CrossRefGoogle Scholar
27.De Filippis, V., An Engel condition with generalized derivations on multilinear polynomials, Israel J. Math. 162 (2007), 93–108.10.1007/s11856-007-0090-yCrossRefGoogle Scholar
28.De Filippis, V., Generalized derivations with Engel condition on multilinear polynomials, Israel J. Math. 171 (2009), 325–348.CrossRefGoogle Scholar
29.Divinsky, N., On commuting automorphisms of rings, Trans. Roy. Soc. Canada. Sect. III 49 (1955), 19–22.Google Scholar
30.Du, Y. and Wang, Y., k-commuting maps on triangular algebras, Linear Algebra Appl. 436 (2012), 1367–1375.10.1016/j.laa.2011.08.024CrossRefGoogle Scholar
31.Faith, C. and Utumi, Y., On Noetherian prime rings, Trans. Amer. Math. Soc. 114 (1965), 53–60.CrossRefGoogle Scholar
32.Franca, W., Commuting maps on some subsets of matrices that are not closed under addition, Linear Algebra Appl. 437 (2012), 388–391.10.1016/j.laa.2012.02.018CrossRefGoogle Scholar
33.Herstein, I. N., Noncommutative rings, Carus Mathematical Monographs, Volume 15 (American Mathematical Society, RI, 1968).Google Scholar
34.Herstein, I. N., Rings with involution (University of Chicago Press, Chicago, 1976).Google Scholar
35.Inceboz, H. G., Koşan, M.-T. and Lee, T.-K., m-power commuting maps on semi-prime rings, Commun. Algebra 42 (2014), 1095–1110.10.1080/00927872.2012.731623CrossRefGoogle Scholar
36.Jacobson, N., Lie algebras (Wisely, New York, 1962).Google Scholar
37.Kissin, E. and Shulman, V. S., Range-inclusive maps on C*-algebras, Q. J. Math. 53 (2002), 455–465.CrossRefGoogle Scholar
38.Lanski, C., Differential identities, Lie ideals, and Posner's theorems, Pacific J. Math. 56 (1986), 231–246.Google Scholar
39.Lanski, C., An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75–80.10.1090/S0002-9939-1993-1132851-9CrossRefGoogle Scholar
40.Lanski, C., An Engel condition with derivation for left ideals, Proc. Amer. Math. Soc. 125 (1997), 339–345.10.1090/S0002-9939-97-03673-3CrossRefGoogle Scholar
41.Lanski, C., Skew derivations and Engel conditions, Commun. Algebra 42 (2014), 139–152.CrossRefGoogle Scholar
42.Lee, P.-H. and Lee, T.-K., Lie ideals of prime rings with derivations, Bull. Inst. Math. Acad. Sinica 11 (1983), 75–80.Google Scholar
43.Lee, P.-H. and Lee, T.-K., Derivations centralizing symmetric or skew elements, Bull. Inst. Math. Acad. Sin. 14 (1986), 249–256.Google Scholar
44.Lee, P.-H. and Lee, T.-K., Linear identities and commuting maps in rings with involution, Commun. Algebra 25 (1997), 2881–2895.CrossRefGoogle Scholar
45.Lee, P.-H. and Wong, T.-L., Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sin. 23 (1995), 1–5.Google Scholar
46.Lee, P.-H. and Wang, Y., Supercentralizing maps in prime superalgebras, Commun. Algebra 37 (2009), 840–854.10.1080/00927870802271672CrossRefGoogle Scholar
47.Lee, T.-K., Semiprime rings with hypercentral derivations, Canad. Math. Bull. 38 (1995), 445–449.CrossRefGoogle Scholar
48.Lee, T.-K., Anti-automorphisms satisfying an Engel condition, Commun. Algebra 45 (2017), 4030–4036.CrossRefGoogle Scholar
49.Lee, T.-K., Commuting anti-homomorphisms, Commun. Algebra 46 (2018), 1060–1065.10.1080/00927872.2017.1335746CrossRefGoogle Scholar
50.Lee, T.-K. and Lee, T.-C., Commuting additive mappings in semiprime rings, Bull. Inst. Math. Acad. Sinica 24 (1996), 259–268.Google Scholar
51.Lee, T.-K., Liu, K.-S. and Shiue, W.-K., n-Commuting maps on prime rings, Publ. Math. Debrecen 63 (2004), 463–857.Google Scholar
52.Liau, P.-K. and Liu, C.-K., An Engel condition with b-generalized derivations for Lie ideals, J. Algebra Appl. 17 (2018), 1850046.10.1142/S0219498818500469CrossRefGoogle Scholar
53.Liu, C.-K., An Engel condition with automorphisms for left ideals, J. Algebra Appl. 13 (2014), 1350092.CrossRefGoogle Scholar
54.Liu, C.-K., An Engel condition with b-generalized derivations, Linear Multilinear Algebra 65 (2017), 300–312.10.1080/03081087.2016.1183560CrossRefGoogle Scholar
55.Liu, C.-K. and Yang, J.-J., Power commuting additive maps on invertible or singular matrices, Linear Algebra Appl. 530 (2017), 127–149.10.1016/j.laa.2017.04.038CrossRefGoogle Scholar
56.Mayne, J. H., Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), 113–115.CrossRefGoogle Scholar
57.Miers, C. R., Centralizing mappings of operator algebras, J. Algebra 59 (1979), 56–64.10.1016/0021-8693(79)90152-2CrossRefGoogle Scholar
58.Posner, E. C., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.CrossRefGoogle Scholar
59.Posner, E. C., Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), 180–183.10.1090/S0002-9939-1960-0111765-5CrossRefGoogle Scholar
60.Qi, X. and Hou, J., k-Commuting additive maps on rings, Linear Algebra Appl. 468 (2015), 48–62.CrossRefGoogle Scholar
61.Rowen, L. H., Polynomial identities in ring theory, Pure and Applied Mathematics, Volume 84 (Academic Press, 1980).Google Scholar
62.Vukman, J., Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), 47–52.CrossRefGoogle Scholar
63.Vukman, J., On derivations in prime rings and Banach algebras, Proc. Amer. Math. Soc. 116 (1992), 877–884.CrossRefGoogle Scholar
64.Xiao, Z. and Wei, F., Commuting mappings of generalized matrix algebras, Linear Algebra Appl. 433 (2010), 2178–2197.10.1016/j.laa.2010.08.002CrossRefGoogle Scholar