 Radical of a module

The radical $J(M)$ of a module is the intersection of its maximal submodules. For a finite-dimensional module over a finite-dimensional algebra $A$,and it is the smallest submodule $N$ for which $M/N$ is a semisimple module.

In the context of modules over a ring, the term "radical" can refer to several concepts, but one common interpretation is the **Jacobson radical** of a module. The Jacobson radical has important implications for the structure and properties of a module. ### Jacobson Radical The Jacobson radical \( \text{Rad}(M) \) of a module \( M \) over a ring \( R \) is defined as the intersection of all maximal submodules of \( M \).