Right ideal

A right ideal $I$ in a unital associative algebra $A$ is a vector subspace with $IA\subseteq I$, equivalently a submodule of the right regular module. It need not be a left ideal. Its finite generation refers to right multiplication by elements of $A$.

A proper right ideal $L$ is maximal if no proper right ideal strictly contains it. Equivalently, $A/L$ is a simple module. The Jacobson radical is the intersection of all maximal right ideals.