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 | References | Similar Articles | Additional Information
Abstract: A functor $T:\mathcal {A} \to \mathcal {B}$ acts faithfully on the right of a class of objects $\mathcal {A}β$ of $\mathcal {A}$ if it distinguishes morphisms out of objects of $\mathcal {A}β$ (that is, $Aβ \in \mathcal {A}β,X \in \mathcal {A},f,g \in \mathcal {A}(Aβ,X)$ 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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Additional Information
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