mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, an essential monomorphism is a monomorphism ''f'' in a category ''C'' such that for a morphism ''g'' in ''C'', the morphism

g \circ f

is a monomorphism only when ''g'' is a monomorphism. Essential monomorphisms in a

category of modules In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring o ...

are those whose image is an

essential submodule In mathematics, specifically module theory, given a ring ''R'' and an ''R''- module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ' ...

of the codomain. An injective hull of an object ''X'' is an essential monomorphism from ''X'' to an injective object.

References

{{cattheory-stub Category theory