Selfinjective

Selfinjective refers to a specific type of algebraic structure, most commonly encountered in ring theory and module theory. A module is called selfinjective if it is isomorphic to a direct summand of its own injective hull. Equivalently, a module is selfinjective if it is injective relative to the class of all modules over its endomorphism ring.

In the context of rings, a ring R is called left selfinjective if the left R-module R itself is selfinjective. Similarly, a ring is right selfinjective if the right R-module R is selfinjective. If a ring is both left and right selfinjective, it is simply called selfinjective.

Selfinjective modules and rings possess several important properties. For instance, if a module is selfinjective and finitely generated, then it is a direct summand of any module containing it as an essential submodule. Selfinjective rings have connections to various other ring constructions and properties, such as Frobenius algebras and Morita duality.

Examples of selfinjective rings include division rings, matrix rings over division rings, and certain group algebras. The study of selfinjective modules and rings is a significant area within abstract algebra, providing insights into the structure and behavior of modules and the rings over which they are defined. Understanding selfinjectivity is crucial for advancing theories related to module decomposition, homological algebra, and the classification of algebraic structures.