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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generating and cogenerating structures


Author: John A. Beachy
Journal: Trans. Amer. Math. Soc. 158 (1971), 75-92
MSC: Primary 18.10
MathSciNet review: 0288160
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Abstract: A functor $ T:\mathcal{A} \to \mathcal{B}$ acts faithfully on the right of a class of objects $ \mathcal{A}'$ of $ \mathcal{A}$ if it distinguishes morphisms out of objects of $ \mathcal{A}'$ (that is, $ A' \in \mathcal{A}',X \in \mathcal{A},f,g \in \mathcal{A}(A',X)$ and $ f \ne g$ implies $ T(f) \ne T(g))$. We define a full subcategory $ \mathcal{R}\mathcal{F}(T)$ such that $ T$ acts faithfully on the right of the objects of $ \mathcal{R}\mathcal{F}(T)$. An object $ U \in \mathcal{A}$ is a generator if $ {H^U}:\mathcal{A} \to \mathcal{E}ns$ is faithful, and if $ {H^U}$ is not faithful, we may still consider $ \mathcal{R}\mathcal{F}({H^U})$. This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of $ R$-modules. Relationships between these can be used in the homological classification of rings.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0288160-3
PII: S 0002-9947(1971)0288160-3
Keywords: $ T$-faithful subcategory, idempotent preradical, radical, generating structure, cogenerating structure, fully divisible $ R$-module, cofaithful $ R$-module, torsionless $ R$-module, faithful $ R$-module, cogenerator ring, $ S$-ring
Article copyright: © Copyright 1971 American Mathematical Society