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Applying the density theorem for derivations to range inclusion problems

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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.
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Bibliografia
  • [1] B. Aupetit, A Primer on Spectral Theory, Springer, 1991.
  • [2] K. I. Beidar and M. Brešar, Extended Jacobson density theorem for rings with derivations and automorphisms, submitted.
  • [3] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, 1996.
  • [4] M. Brešar, Derivations on noncommutative Banach algebras II, Arch. Math. (Basel) 61 (1994), 56-59.
  • [5] M. Brešar, Derivations mapping into the socle, II, Proc. Amer. Math. Soc. 126 (1998), 181-188.
  • [6] M. Brešar and P. Šemrl, On locally linearly dependent operators and derivations, Trans. Amer. Math. Soc. 351 (1999), 1257-1275.
  • [7] L. O. Chung and J. Luh, Nilpotency of derivations, Canad. Math. Bull. 26 (1983), 341-346.
  • [8] B. Felzenszwalb and C. Lanski, On the centralizers of ideals and nil derivations, J. Algebra 83 (1983), 520-530.
  • [9] I. N. Herstein, A note on derivations, Canad. Math. Bull. 21 (1978), 369-370.
  • [10] I. N. Herstein, Sui commutatori degli anelli semplici, Rend. Sem. Mat. Fis. Milano 33 (1963), 80-86.
  • [11] V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic 17 (1978), 155-168.
  • [12] C. Lanski, Derivations nilpotent on subsets of prime rings, Comm. Algebra 20 (1992), 1427-1446.
  • [13] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 731-734.
  • [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.
  • [15] W. S. Martindale III and C. R. Miers, On the iterates of derivations of prime rings, Pacific J. Math. 104 (1983), 179-190.
  • [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.
  • [17] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474.
  • [18] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487.
  • [19] G. J. Murphy, Aspects of the theory of derivations, in: Banach Center Publ. 30 1994, 267-275.
  • [20] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211.
  • [21] V. Runde, Range inclusion results for derivations on noncommutative Banach algebras, Studia Math. 105 (1993), 159-172.
  • [22] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170.
  • [23] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264.
  • [24] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460.
  • [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).
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