selfinjectivity

Self-injectivity is a concept primarily found in abstract algebra, specifically within the study of modules over a ring. A module M over a ring R is called self-injective if every R-homomorphism from a submodule of M to M can be extended to an R-homomorphism from M to M itself. In simpler terms, any R-linear map that "works" on a piece of the module can be "completed" to work on the entire module.

The notion of self-injectivity is closely related to the concept of injective modules. An injective module is a module that is self-injective with respect to all modules. However, the definition of self-injective is more specific, referring to homomorphisms that map into the module itself, rather than from any arbitrary module.

Self-injective modules play a significant role in various areas of algebra. For instance, in ring theory, a ring is left self-injective if it is self-injective as a left module over itself. These rings have special properties and are related to concepts like Baer rings and $C^$-algebras. The study of self-injective modules helps in understanding the structure of modules and rings, particularly in decomposition theorems and homological algebra. Examples of self-injective modules include injective hulls, which are essential for embedding modules into larger injective structures. The existence and properties of self-injective modules are crucial for constructing resolutions and studying homological invariants.