The problem of when derivations (and their powers) have the range in the Jacobson radical is considered. The proofs are based on the density theorem for derivations.
@article{bwmeta1.element.bwnjournal-article-smv138i1p93bwm,
author = {K. Beidar and Matej Bre\v sar},
title = {Applying the density theorem for derivations to range inclusion problems},
journal = {Studia Mathematica},
volume = {141},
year = {2000},
pages = {93-100},
zbl = {0954.16026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv138i1p93bwm}
}
Beidar, K.; Brešar, Matej. Applying the density theorem for derivations to range inclusion problems. Studia Mathematica, Tome 141 (2000) pp. 93-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv138i1p93bwm/
[00000] [1] B. Aupetit, A Primer on Spectral Theory, Springer, 1991.
[00001] [2] K. I. Beidar and M. Brešar, Extended Jacobson density theorem for rings with derivations and automorphisms, submitted. | Zbl 1003.47029
[00002] [3] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, 1996.
[00003] [4] M. Brešar, Derivations on noncommutative Banach algebras II, Arch. Math. (Basel) 61 (1994), 56-59.
[00004] [5] M. Brešar, Derivations mapping into the socle, II, Proc. Amer. Math. Soc. 126 (1998), 181-188.
[00005] [6] M. Brešar and P. Šemrl, On locally linearly dependent operators and derivations, Trans. Amer. Math. Soc. 351 (1999), 1257-1275. | Zbl 0920.15009
[00006] [7] L. O. Chung and J. Luh, Nilpotency of derivations, Canad. Math. Bull. 26 (1983), 341-346. | Zbl 0476.16028
[00007] [8] B. Felzenszwalb and C. Lanski, On the centralizers of ideals and nil derivations, J. Algebra 83 (1983), 520-530. | Zbl 0519.16022
[00008] [9] I. N. Herstein, A note on derivations, Canad. Math. Bull. 21 (1978), 369-370. | Zbl 0412.16018
[00009] [10] I. N. Herstein, Sui commutatori degli anelli semplici, Rend. Sem. Mat. Fis. Milano 33 (1963), 80-86.
[00010] [11] V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic 17 (1978), 155-168. | Zbl 0423.16011
[00011] [12] C. Lanski, Derivations nilpotent on subsets of prime rings, Comm. Algebra 20 (1992), 1427-1446. | Zbl 0759.16014
[00012] [13] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 731-734. | Zbl 0821.16037
[00013] [14] C. Le Page, Sur quelques conditions entraȋnant la commutativité dans les algèbres de Banach, C. R. Acad. Sci. Paris Sér. A 265 (1967), 235-237. | Zbl 0158.14102
[00014] [15] W. S. Martindale III and C. R. Miers, On the iterates of derivations of prime rings, Pacific J. Math. 104 (1983), 179-190. | Zbl 0507.16022
[00015] [16] M. Mathieu, Where to find the image of a derivation, in: Banach Center Publ. 30, Inst. Math. Polish Acad. Sci., Warszawa, 1994, 237-249. | Zbl 0813.47043
[00016] [17] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474. | Zbl 0714.46038
[00017] [18] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487. | Zbl 0733.46023
[00018] [19] G. J. Murphy, Aspects of the theory of derivations, in: Banach Center Publ. 30 1994, 267-275. | Zbl 0811.46045
[00019] [20] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211. | Zbl 0407.46043
[00020] [21] V. Runde, Range inclusion results for derivations on noncommutative Banach algebras, Studia Math. 105 (1993), 159-172. | Zbl 0810.46044
[00021] [22] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170.
[00022] [23] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. | Zbl 0067.35101
[00023] [24] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460. | Zbl 0681.47016
[00024] [25] Yu. V. Turovskiĭ and V. S. Shul'man, Conditions for the massiveness of the range of a derivation of a Banach algebra and of associated differential operators, Mat. Zametki 42 (1987), 305-314 (in Russian).