Jacobson radical

The Jacobson radical is a concept in algebra that identifies parts of a ring (a mathematical structure) related to elements that act almost like zero in certain contexts. Specifically, it consists of elements that, when combined with any other element, make the whole system behave like it has no significant "invertible" elements—those you can "undo" or reverse. Think of it as capturing elements that are "nearly negligible" in the structure, helping mathematicians understand how the ring behaves and how it can be broken down into simpler components for analysis.