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  • 1 Alexandria University, Alexandria, Egypt
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The consistent way of investigating rings with involution, briefly *-rings, is to study them in the category of *-rings with morphisms preserving also involution. In this paper we continue the study of *-rings and the notion of *-reduced *-rings is introduced and their properties are studied. We introduce also the class of *-Baer *-rings. This class is defined in terms of *-annihilators and principal *-biideals, and it naturally extends the class of Baer *-rings. The use of *-biideals makes this concept more consistent with the involution than the use of right ideals in the notion of Baer *-rings. We prove that each *-Baer *-ring is semiprime. Moreover, we show that the property of *-Baer extends to both the *-corner and the center of the *-ring. Finally, we discuss the relation between *-Baer and quasi-Baer *-rings; the generalization of Baer *-ring.

  • 1

    Aburawash, U. A., Usama A., Semiprime involution rings and chain conditions, Contr. to General Algebra 7, Hölder—Pichler—Tempsky, Wien and B. G. Teubner, Stuttgart (1991), 711.

    • Search Google Scholar
    • Export Citation
  • 2

    Aburawash, U. A., Usama A., On involution rings, East-West J. Math., 2(2) (2000), 109126.

  • 3

    Aburawash, U. A. and Sola, K. B., *-Zero divisors and *-prime ideals, East-West J. Math., 12(1) (2010), 2731.

  • 4

    Beidar, K. I. and Wiegandt, R., Rings with involution and chain conditions, J. Pure and Appl. Algebra, 87 (1993), 205220.

  • 5

    Beidar, K. I., Márki, L., Mlitz, R. and Wiegandt, R., Primitive involution rings, Acta Math. Hungar., 109 (2005), 357368.

  • 6

    Berberian, S. K., Baer *-Rings, Grundlehren Math. Wiss. Band. 195, Springer-Verlag, Berlin–Heidelberg–New York, 1972.

  • 7

    Berberian, S. K., Baer Rings and Baer *-Rings, The University of Texas at Austin, March 1988.

  • 8

    Birkenmeier, G. F. and Groenewald, N. J., Prime ideals in rings with involution, Quaestiones in Mathematicae, 20 (1997), 591603.

  • 9

    Birkenmeier, G. F. and Park, J. K., Self-adjoint ideals in Baer *-ring, Comm. Algebra, 28 (2000), 42594268.

  • 10

    Birkenmeier, G. F., Kim, J. Y. and Park, J. K., On Quasi-Baer rings, Contemporary Mathematics, 259 (2000), 6792.

  • 11

    Birkenmeier, G. F., Kim, J. Y. and Park, J. K., Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra, 159 (2001), 2542.

    • Search Google Scholar
    • Export Citation
  • 12

    Birkenmeier, G. F., Park, J. K. and Rizvi, S. T., Hulls of semiprime rings with applications to C*-Algebra, J. Algebra, 322 (2009), 327352.

    • Search Google Scholar
    • Export Citation
  • 13

    Herstein, I. N., Rings with Involution, Univ. Chicago Press, 1976.

  • 14

    Kaplansky, I., Rings of Operators, Benjamin, New York, 1968.

  • 15

    Rowen, L. H., Ring Theory I, Academic Press, San Diego, 1988.

  • 16

    Lee, Y., Kim, N. K. and Hong, C. Y., Counterexamples on Baer rings, Comm. Algebra, 25 (1997), 497507.