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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
DOI: https://doi.org/10.1090/S0002-9947-1971-0288160-3
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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Keywords: <IMG WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img5.gif" ALT="$T$">-faithful subcategory, idempotent preradical, radical, generating structure, cogenerating structure, fully divisible <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$R$">-module, cofaithful <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$R$">-module, torsionless <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$R$">-module, faithful <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img21.gif" ALT="$R$">-module, cogenerator ring, <IMG WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$S$">-ring
Article copyright: © Copyright 1971 American Mathematical Society