A Survey on Free Objects in Division Rings and in Division Rings with an Involution

REFERENCES

• Amitsur , S. A. ( 1966 ). Rational identities and applications to algebra and geometry . J. Algebra 3 : 304 – 359 .
   Web of Science ®Google Scholar
• Chiba , K. ( 1996 ). Free subgroups and free subsemigroups in division rings . J. Algebra 184 : 570 – 574 .
   Web of Science ®Google Scholar
• Ferreira , V. O. , Goncalves , J. Z. , Mandel , A. ( 2005 ). Free symmetric and unitary pairs in division rings with involution . Internat. J. Algebra Comp. 15 ( 1 ): 15 – 36 .
   Google Scholar
• Figueiredo , L. M. V. , Goncalves , J. Z. , Shirvani , M. ( 1996 ). Free group algebras in certain division rings . J. Algebra 185 : 298 – 313 .
   Web of Science ®Google Scholar
• Goncalves , J. Z. ( 1984 ). Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings . Canad. Math. Bull. 27 : 365 – 370 .
   Web of Science ®Google Scholar
• Goncalves , J. Z. , Mandel , A. ( 1986 ). Are there free groups in division rings? Israel J. Math. 53 ( 1 ): 69 – 80 .
   Google Scholar
• Goncalves , J. Z. , Passman , D. S. ( 2003 ). Free unit groups in algebras . Comm. Algebra 31 ( 5 ): 2219 – 2227 .
   Web of Science ®Google Scholar
• Goncalves , J. Z. , Passman , D. S. ( 2006 ). Linear groups and group rings . J. Algebra 295 : 94 – 118 .
   Google Scholar
• Goncalves , J. Z. , Mandel , A. , Shirvani , M. ( 1999 ). Free products of units in algebras I. Quaternion algebras . J. Algebra 214 : 301 – 316 .
   Web of Science ®Google Scholar
• Goncalves , J. Z. , Mandel , A. , Shirvani , M. ( 2000 ). Free products of units in algebras. II. Crossed products . J. Algebra 233 : 567 – 593 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 1996 ). On free group algebras in division rings with uncountable center . Proceedings AMS 124 ( 3 ): 685 – 687 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 1998 ). Free group algebras in the field of fractions of differential polynomial rings and enveloping algebras . J. Algebra 204 : 372 – 385 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 1999 ). Large free algebras in the ring of fractions of skew polynomial rings . J. London Math. Soc. 2 ( 60 ): 481 – 489 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 2005 ). Free products arising from elements of finite order in simple rings . Proceedings AMS 133 ( 7 ): 1919 – 1923 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 2006 ). Free symmetric and unitary pairs in central simple algebras with involution . Contemporary Math. 420 : 121 – 139 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 2009 ). Algebraic elements as free factors in simple artinian rings . Contemporary Math. 499 : 121 – 125 .
   Google Scholar
• Goncalves , J. Z. , Shirvani , M. ( 2008 ). Free groups in central simple algebras without Tits’ theorem . Comm. Algebra 36 : 3113 – 3121 .
   Web of Science ®Google Scholar
• Goncalves , J. Z. , Tengan , E. ( 2009 ). A note on free groups in the ring of fractions of skew polynomial rings . Comm. Algebra 37 : 2477 – 2484 .
   Web of Science ®Google Scholar
• Hamilton , W. R. ( 1853 ). Lectures on Quaternions . Dublin .
   Google Scholar
• Herstein , I. N. ( 1978 ). Multiplicative commutators in division rings . Israel J. Math. 31 : 180 – 188 .
   Web of Science ®Google Scholar
• Hilbert , D. ( 1930 ). Grundlagen der Geometrie . Leipzig , Berlin : Teubner .
   Google Scholar
• Hua , L.-K. ( 1950 ). On the multiplicative group of a sfield . Acad. Sinica Sci. Record 3 : 1 – 6 .
   Google Scholar
• Jacobson , N. ( 1945 ). Structure theory of algebraic algebras of bounded degree . Annals Math. 46 : 695 – 707 .
   Web of Science ®Google Scholar
• Kaplansky , I. (1948). Rings with polynomial identity. Bull. Amer. Math. Soc. 54:575–580.
   Web of Science ®Google Scholar
• Kaplansky , I. ( 1951 ). A theorem on division rings . Canadian J. Math. 3 : 290 – 292 .
   Google Scholar
• Lam , T. Y. ( 1991 ). A First Course in Noncommutative Rings . GTM 131 . New York : Springer-Verlag .
   Google Scholar
• Lichtman , A. I. ( 1977 ) On the subgroups of the multiplicative group of skewfields . Proceedings A.M.S. 63 : 15 – 16 .
   Web of Science ®Google Scholar
• Lichtman , A. I. ( 1978 ). Free subgroups of normal subgroups of the multiplicative group of skewfields . Proceedings A. M. S. 71 : 174 – 178 .
   Web of Science ®Google Scholar
• Lichtman , A. I. ( 1982 ). On normal subgroups of multiplicative groups of skewfields generated by polycyclic-by-finite groups . J. Algebra 78 : 548 – 577 .
   Web of Science ®Google Scholar
• Lichtman , A. I. ( 1984 ). On matrix rings and linear groups over fields of fractions of group rings and enveloping algebras, II . J. Algebra 90 : 516 – 527 .
   Web of Science ®Google Scholar
• Lichtman , A. I. ( 1999 ). Free subalgebras in division rings generated by universal enveloping algebras . Algebra Coll. 6 ( 2 ): 145 – 153 .
   Web of Science ®Google Scholar
• Lorenz , M. ( 1984 ). Group Rings and Division Rings . Methods in Ring Theory. Reidel, Dordrecht , pp. 265 – 280 .
   Google Scholar
• Lorenz , M. ( 1986 ). On free subalgebras of certain division rings . Proceedings AMS 98 : 401 – 405 .
   Web of Science ®Google Scholar
• Makar-Limanov , L. ( 1983 ). The skewfield of fractions of the Weyl algebra contains free noncommutative algebra . Comm. Algebra 17 : 2003 – 2006 .
   Google Scholar
• Makar-Limanov , L. ( 1984 ). On the group rings of nilpotent groups . Israel J. Math. 48 : 245 – 248 .
   Google Scholar
• Makar-Limanov , L. ( 1984 ). On Free Subobjects of Skewfields . Methods in Ring Theory. NATO ASI, Series C, 233. Reidel, Boston , pp. 281 – 285 .
   Google Scholar
• Makar-Limanov , L. ( 1991 ). On subalgebras of the first Weyl skewfield . Comm. Algebra 19 : 1971 – 1982 .
   Google Scholar
• Makar-Limanov , L. , Malcolmson , P. ( 1991 ). Free subalgebras of enveloping algebras . Proceedings AMS 111 : 315 – 322 .
   Web of Science ®Google Scholar
• Passman , D. S. ( 1982 ). Universal field of fractions for polycyclic group algebras . Glasgow Math. J. 23 : 103 – 113 .
   Google Scholar
• Reichstein , Z. ( 1996 ). On a question of Makar-Limanov . Proceedings AMS 124 ( 1 ): 17 – 19 .
   Google Scholar
• Scott , W.R. ( 1987 ). Group Theory . New York : Dover .
   Google Scholar
• Shorey , T. , Tijdman , R. Exponential Diophantine Equations . Cambridge : Cambridge U.P.
   Google Scholar
• Smoktunowicz , A. ( 2009 ). Makar-Limanov's conjecture on free subalgebras . Advances Math. 222 ( 6 ): 2007 – 2016 .
   Google Scholar
• Tits , J. ( 1972 ). Free groups in linear groups . J. Algebra 20 : 250 – 270 .
   Web of Science ®Google Scholar
• Wedderburn , J. H. M. ( 1905 ). A theorem on finite algebras . Trans. American Math. Soc. 6 : 349 – 352 .
   Google Scholar
• Communicated by S. Sehgal.
   Google Scholar