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If a ring is injective over itself as a right module, then it is called a right self-injective ring. en.wikipedia.org

The map between these quotient groups is either an isomorphism or is injective and has an image of index 2. en.wikipedia.org

The converse is a result of: if every module has a unique maximal injective submodule, then the ring is hereditary. en.wikipedia.org

When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion. en.wikipedia.org

Given a finite-dimensional module over a commutative ring, the pairing may be injective (hence nondegenerate in the above sense) but not unimodular. en.wikipedia.org

The category of abelian sheaves has enough injective objects: this means that any sheaf is a subsheaf of an injective sheaf. en.wikipedia.org

There is an analogous definition using injective resolutions and cochain complexes. en.wikipedia.org

The function "f" itself need not be injective, only its derivative. en.wikipedia.org

In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves. en.wikipedia.org

In fact, injective sheaves are flabby ("flasque"), soft, and acyclic. en.wikipedia.org

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