Many authors in abstract algebra and universal algebra define an "'epimorphism "'simply as an " onto " or surjective homomorphism.
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As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context.
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_{ R } M is a semiartinian module if, for all M \ rightarrow N epimorphism, where N \ neq0, the socle of N is essential in N.
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Accordingly, "'right quasi-abelian categories "'are pre-abelian categories such that \ overline { f } is an epimorphism for each morphism f.
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The companion terms " monomorphism " and " epimorphism " were originally introduced by Nicolas Bourbaki; Bourbaki uses " monomorphism " as shorthand for an injective function.
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Other examples come from the fact that finite Hopf-Galois extensions are depth two in a strong sense ( the split epimorphism in the definition may be replaced by a bimodule isomorphism ).
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Many common notions from mathematics ( e . g . surjective, injective, free object, representation, isomorphism ) are definable purely in category theoretic terms ( cf . monomorphism, epimorphism ).
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An extremal monomorphism is a monomorphism that cannot be nontrivially factored through an epimorphism : Precisely, if with " e " an epimorphism, then " e " is an isomorphism.
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An extremal monomorphism is a monomorphism that cannot be nontrivially factored through an epimorphism : Precisely, if with " e " an epimorphism, then " e " is an isomorphism.
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The categorical dual of a monomorphism is an epimorphism, i . e . a monomorphism in a category " C " is an epimorphism in the dual category " C " op.
