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Projective Covers and Perfect Rings

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Algebra II Ring Theory

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 191))

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Abstract

A morphism f: A → B of R-modules is said to be minimal provided that ker f is a superfluous submodule of A. For example, for a right ideal I, the canonical map R → R/I is superfluous if and only if I ⊆ rad R 18.3. A module A is a projective cover (proj. cov.) of B provided that A is projective and there exists a minimal epimorphism A → B. This notion is dual to that of injective hull, and yet, although each R-module has an injective hull, projective covers of modules may fail to exist. For example, as is shown in this chapter, a necessary condition that every right R-module have a projective cover is that R/rad R be semisimple, and rad R be a nil ideal.

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Authors and Affiliations

  1. Rutgers, The State University, 08903, New Brunswick, N.J., USA

    Carl Faith

  2. The Institute for Advanced Study, 08540, Princeton, N.J., USA

    Carl Faith

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  1. Carl Faith
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Editors and Affiliations

  1. Rutgers, The State University, 08903, New Brunswick, N.J., USA

    Carl Faith

  2. The Institute for Advanced Study, 08540, Princeton, N.J., USA

    Carl Faith

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© 1976 Springer-Verlag Berlin Heidelberg

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Faith, C. (1976). Projective Covers and Perfect Rings. In: Faith, C. (eds) Algebra II Ring Theory. Grundlehren der mathematischen Wissenschaften, vol 191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65321-6_7

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  • DOI: https://doi.org/10.1007/978-3-642-65321-6_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-65323-0

  • Online ISBN: 978-3-642-65321-6

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