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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Right orders in full linear rings


Author: K. C. O’Meara
Journal: Trans. Amer. Math. Soc. 203 (1975), 299-318
MSC: Primary 16A18
DOI: https://doi.org/10.1090/S0002-9947-1975-0360663-6
MathSciNet review: 0360663
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Abstract: In this paper a right order $ R$ in an infinite dimensional full linear ring is characterized as a ring satisfying:

(1) $ R$ is meet-irreducible (with zero right singular ideal) and contains uniform right ideals;

(2) the closed right ideals of $ R$ are right annihilator ideals, and each such right ideal is essentially finitely generated;

(3) $ R$ possesses a reducing pair (i.e. a pair $ ({\beta _1},{\beta _2})$ of elements for which $ {\beta _1}R,{\beta _2}R$ and $ \beta _1^r + \beta _2^r$ are large right ideals of $ R$);

(4) for each $ a \in R$ with $ {a^l} = 0,aR$ contains a regular element of $ R$.

A second characterization of $ R$ is also given. This is in terms of the right annihilator ideals of $ R$ which have the same (uniform) dimension as $ {R_R}$.

The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.


References [Enhancements On Off] (What's this?)

  • [1] V. C. Cateforis, Flat regular quotient rings, Trans. Amer. Math. Soc. 138 (1969), 241-249. MR 39 #259. MR 0238899 (39:259)
  • [2] -, On regular self-injective rings, Pacific J. Math. 30 (1969), 39-45. MR 40 #1432. MR 0248178 (40:1432)
  • [3] C. Faith, Lectures on injective modules and quotient rings, Lecture Notes in Math., no. 49, Springer-Verlag, Berlin and New York, 1967. MR 37 #2791. MR 0227206 (37:2791)
  • [4] G. D. Findlay, Flat epimorphic extensions of rings, Math. Z. 118 (1970), 281-288. MR 43 #6270. MR 0280550 (43:6270)
  • [5] A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201-220. MR 22 #2627. MR 0111766 (22:2627)
  • [6] J. J. Hutchinson, Quotient full linear rings, Proc. Amer. Math. Soc. 28 (1971), 375-378. MR 0424867 (54:12825)
  • [7] R. E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895. MR 13, 618. MR 0045695 (13:618c)
  • [8] -, Structure theory of faithful rings. III. Irreducible rings, Proc. Amer. Math. Soc. 11 (1960), 710-717. MR 22 #9510. MR 0118739 (22:9510)
  • [9] -, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11 (1961), 1385-1392. MR 26 #1331. MR 0143779 (26:1331)
  • [10] -, Rings of finite rank, Publ. Math. Debrecen 11 (1964), 284-287. MR 31 #3454. MR 0179205 (31:3454)
  • [11] J. Lambek, Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966. MR 34 #5857. MR 0206032 (34:5857)
  • [12] Y. Miyashita, On quasi-injective modules. A generalization of the theory of completely reducible modules, J. Fac. Sci. Hokkaido Univ. Ser. I 18 (1965), 158-187. MR 30 #2044. MR 0171817 (30:2044)
  • [13] K. Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971), 25-40. MR 44 #4040. MR 0286833 (44:4040)
  • [14] K. C. O'Meara, Primeness of right orders in full linear rings, J. Algebra 26 (1973), 172-184. MR 0332857 (48:11182)
  • [15] -, Intrinsic extensions of prime rings, Pacific J. Math. 51 (1974), 257-269. MR 0480608 (58:764)
  • [16] N. Popescu and T. Spircu, Quelques observations sur les èpimorphismes plats (à gauche) d'anneaux, J. Algebra 16 (1970), 40-59. MR 42 #323. MR 0265413 (42:323)
  • [17] N. Popescu and D. Spulber, Sur les quasi-ordres (à gauche) dans un anneau, J. Algebra 17 (1971), 474-481. MR 43 #282. MR 0274519 (43:282)
  • [18] L. Silver, Noncommutative localizations and applications, J. Algebra 7 (1967), 44-76. MR 36 #205. MR 0217114 (36:205)
  • [19] Y. Utumi, On rings of which any one-sided quotient rings are two-sided, Proc. Amer. Math. Soc. 14 (1963), 141-147. MR 26 #137. MR 0142568 (26:137)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0360663-6
Keywords: Full linear ring, right order, right quasi-order, irreducible ring, uniform right ideal, uniform dimension, closed right ideal, right annihilator ideal, essentially finitely generated, reducing pair, flat epimorphic extension, idealizer
Article copyright: © Copyright 1975 American Mathematical Society

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