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Transactions of the American Mathematical Society

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

 
 

 

$ I$-rings


Author: W. K. Nicholson
Journal: Trans. Amer. Math. Soc. 207 (1975), 361-373
MSC: Primary 16A32
DOI: https://doi.org/10.1090/S0002-9947-1975-0379576-9
MathSciNet review: 0379576
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Abstract: A ring $ R$, possibly with no identity, is called an $ {I_0}$-ring if each one-sided ideal not contained in the Jacobson radical $ J(R)$ contains a nonzero idempotent. If, in addition, idempotents can be lifted modulo $ J(R),R$ is called an $ I$-ring. A survey of when these properties are inherited by related rings is given. Maximal idempotents are examined and conditions when $ {I_0}$-rings have an identity are given. It is shown that, in an $ {I_0}$-ring $ R$, primitive idempotents are local and primitive idempotents in $ R/J(R)$ can always be lifted. This yields some characterizations of $ {I_0}$-rings $ R$ such that $ R/J(R)$ is primitive with nonzero socle. A ring $ R$ (possibly with no identity) is called semiperfect if $ R/J(R)$ is semisimple artinian and idempotents can be lifted modulo $ J(R)$. These rings are characterized in several new ways: among them as $ {I_0}$-rings with no infinite orthogonal family of idempotents, and as $ {I_0}$-rings $ R$ with $ R/J(R)$ semisimple artinian. Several other properties are derived. The connection between $ {I_0}$-rings and the notion of a regular module is explored. The rings $ R$ which have a regular module $ M$ such that $ J(R) = \operatorname{ann} (M)$ are studied. In particular they are $ {I_0}$-rings. In addition, it is shown that, over an $ {I_0}$-ring, the endomorphism ring of a regular module is an $ {I_0}$-ring with zero radical.


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  • [1] A. S. Amitsur, A general theory of radicals. II. Radicals in rings and bi-categories, Amer. J. Math. 76 (1954), 100-125. MR 15, 499. MR 0059256 (15:499b)
  • [2] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. MR 28 #1212. MR 0157984 (28:1212)
  • [3] N. Bourbaki, Éléments de mathématique. XXIII. Livre II: Algèbre, Chap. 8: Modules et anneaux semisimple, Actualités Sci. Indust., no. 1261, Hermann, Paris, 1958. MR 20 #4576. MR 0098114 (20:4576)
  • [4] N. Jacobson, Structure of rings, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1964. MR 36 #5158. MR 0222106 (36:5158)
  • [5] I. Kaplansky, Rings of operators, Benjamin, New York, 1968. MR 39 #6092. MR 0244778 (39:6092)
  • [6] K. Koh, On Von Neumann regular rings, Canad. Math. Bull. 17 (1974), 283-284. MR 0357483 (50:9951)
  • [7] J. Krempa, Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), no. 2, 121-130. MR 46 #5377. MR 0306251 (46:5377)
  • [8] J. Levitzki, On the structure of algebraic algebras and related rings, Trans. Amer. Math. Soc. 74 (1953), 384-409. MR 14, 720. MR 0053089 (14:720a)
  • [9] R. Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233-256. MR 43 #274. MR 0274511 (43:274)
  • [10] J. Zelmanowitz, Regular modules, Trans. Amer. Math. Soc. 163 (1972), 341-355. MR 44 #4050. MR 0286843 (44:4050)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0379576-9
Keywords: $ I$-ring, idempotents, lifting idempotents, semiperfect ring, von Neumann regular ring, regular module, endomorphism ring
Article copyright: © Copyright 1975 American Mathematical Society

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