The Jacobson radical is the intersection of all maximal left ideals, equivalently the intersection of the annihilators of all simple left modules.
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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 \( R \).