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Hopfian and co-Hopfian objects.

Kalathoor Varadarajan (1992)

Publicacions Matemàtiques

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...

Immersions of module varieties

Grzegorz Zwara (1999)

Colloquium Mathematicae

We show that a homomorphism of algebras is a categorical epimorphism if and only if all induced morphisms of the associated module varieties are immersions. This enables us to classify all minimal singularities in the subvarieties of modules from homogeneous standard tubes.

Jordan automorphisms of triangular algebras. II

Driss Aiat Hadj Ahmed, Rachid Tribak (2015)

Commentationes Mathematicae Universitatis Carolinae

We give a sufficient condition under which any Jordan automorphism of a triangular algebra is either an automorphism or an anti-automorphism.

Modules which are invariant under idempotents of their envelopes

Le Van Thuyet, Phan Dan, Truong Cong Quynh (2016)

Colloquium Mathematicae

We study the class of modules which are invariant under idempotents of their envelopes. We say that a module M is -idempotent-invariant if there exists an -envelope u : M → X such that for any idempotent g ∈ End(X) there exists an endomorphism f : M → M such that uf = gu. The properties of this class of modules are discussed. We prove that M is -idempotent-invariant if and only if for every decomposition X = ⨁ i ∈ I X i , we have M = ⨁ i ∈ I ( u - 1 ( X i ) ∩ M ) . Moreover, some generalizations of -idempotent-invariant modules are considered....

Monomorphisms of coalgebras

A. L. Agore (2010)

Colloquium Mathematicae

We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, φ: C → D is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras C and D coincide if and only if ∑ i ∈ I ε ( a i ) b i = ∑ i ∈ I a i ε ( b i ) for all ∑ i ∈ I a i ⊗ b i ∈ C ◻ D C . In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

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