epimorphism in a sentence

Examples

1. Many authors in abstract algebra and universal algebra define an "'epimorphism "'simply as an " onto " or surjective homomorphism.
2. 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.
3. _{ R } M is a semiartinian module if, for all M \ rightarrow N epimorphism, where N \ neq0, the socle of N is essential in N.
4. Accordingly, "'right quasi-abelian categories "'are pre-abelian categories such that \ overline { f } is an epimorphism for each morphism f.
5. The companion terms " monomorphism " and " epimorphism " were originally introduced by Nicolas Bourbaki; Bourbaki uses " monomorphism " as shorthand for an injective function.
6. 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 ).
7. Many common notions from mathematics ( e . g . surjective, injective, free object, representation, isomorphism ) are definable purely in category theoretic terms ( cf . monomorphism, epimorphism ).
8. 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.
9. 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.
10. 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.