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Direct Sum Representations of Rings and Modules

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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

If C is a class of right R-modules, a module M will be called sigma C, in case M is isomorphic to a direct sum of modules in C. We will write Σ-C for short. Also, if M is a class of right R-modules which is Σ-C then we say that R is right ℳ Σ-C. For example, if ℳ is the class of (injective) right R-modules, and C is the class of finitely generated right R-modules, then the corresponding statement is that R is right (injective) Σ-finitely generated. Also, the statement R is right (injective) Σ-cyclic means that every (injective) right R-module is a direct sum of cyclic modules. (Note that one applies these designations to R as an object of RINGS rather than as an object of mod-R; strictly speaking, it is mod-R (not R) that is (injective) Σ-finitely generated when we say that R is.)

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  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. 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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Faith, C. (1976). Direct Sum Representations of Rings and Modules. 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_5

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