$I$-rings
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- by W. K. Nicholson PDF
- Trans. Amer. Math. Soc. 207 (1975), 361-373 Request permission
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.References
- S. A. Amitsur, A general theory of radicals. II. Radicals in rings and bicategories, Amer. J. Math. 76 (1954), 100–125. MR 59256, DOI 10.2307/2372403
- Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. MR 157984, DOI 10.1090/S0002-9947-1960-0157984-8
- N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1261, Hermann, Paris, 1958 (French). MR 0098114
- Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
- Irving Kaplansky, Rings of operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0244778
- Kwangil Koh, On von Neumann regular rings, Canad. Math. Bull. 17 (1974), 283–284. MR 357483, DOI 10.4153/CMB-1974-055-1
- Jan Krempa, Logical connections between some open problems concerning nil rings, Fund. Math. 76 (1972), no. 2, 121–130. MR 306251, DOI 10.4064/fm-76-2-121-130
- Jakob Levitzki, On the structure of algebraic algebras and related rings, Trans. Amer. Math. Soc. 74 (1953), 384–409. MR 53089, DOI 10.1090/S0002-9947-1953-0053089-1
- Roger Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256. MR 274511, DOI 10.1090/S0002-9947-1971-0274511-2
- J. Zelmanowitz, Regular modules, Trans. Amer. Math. Soc. 163 (1972), 341–355. MR 286843, DOI 10.1090/S0002-9947-1972-0286843-3
Additional Information
- © Copyright 1975 American Mathematical Society
- 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