Jacobson radical

The Jacobson radical is a concept that arises in the context of ring theory, a branch of abstract algebra. It is a particular ideal associated with a ring, which captures information about the ring's structure in relation to simple modules and semisimplicity. Here are the key points regarding the Jacobson radical: 1. **Definition**: The Jacobson radical \( J(R) \) of a ring \( R \) is defined as the intersection of all maximal left ideals of \( R \).