Hopfian and co-Hopfian objects.

Varadarajan, Kalathoor. "Hopfian and co-Hopfian objects.." Publicacions Matemàtiques 36.1 (1992): 293-317. <http://eudml.org/doc/41149>.

@article{Varadarajan1992,
abstract = {The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space X let CR(X) (resp. CC(X)) denote the R (resp. C)-algebra of real (resp. complex) valued continuous functions on X. Using Gelfand's representation theorem we will prove that CR(X) (CC(X)) is Hopfian (respectively co-Hopfian) as an R(C) algebra if and only if X is co-Hopfian (respectively Hopfian) as an object of Top. We also study Hopfian and co-Hopfian compact topological manifolds.},
author = {Varadarajan, Kalathoor},
journal = {Publicacions Matemàtiques},
keywords = {Anillos; Algebra de Hopf; Espacio de funciones continuas; Módulos; Hopficity; co-Hopficity; co-Hopfian as -module; Boolean ring; Boolean space; category Top; compact Hausdorff space; compact manifold},
language = {eng},
number = {1},
pages = {293-317},
title = {Hopfian and co-Hopfian objects.},
url = {http://eudml.org/doc/41149},
volume = {36},
year = {1992},
}

TY - JOUR
AU - Varadarajan, Kalathoor
TI - Hopfian and co-Hopfian objects.
JO - Publicacions Matemàtiques
PY - 1992
VL - 36
IS - 1
SP - 293
EP - 317
AB - The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space X let CR(X) (resp. CC(X)) denote the R (resp. C)-algebra of real (resp. complex) valued continuous functions on X. Using Gelfand's representation theorem we will prove that CR(X) (CC(X)) is Hopfian (respectively co-Hopfian) as an R(C) algebra if and only if X is co-Hopfian (respectively Hopfian) as an object of Top. We also study Hopfian and co-Hopfian compact topological manifolds.
LA - eng
KW - Anillos; Algebra de Hopf; Espacio de funciones continuas; Módulos; Hopficity; co-Hopficity; co-Hopfian as -module; Boolean ring; Boolean space; category Top; compact Hausdorff space; compact manifold
UR - http://eudml.org/doc/41149
ER -