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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.
@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 -