Jacobson Radical Of An Algebra

[Tsakanikas Nickos]

Let \( A \) be a \( \mathbb{K} \)-algebra, where \( \mathbb{K} \) is a field (we do not assume commutativity for \( A \)). Let \( x \in A \) and let \( J(A) \) be the Jacobson radical of \( A \). Show that the following are equivalent:

1. \( x \in J(A) \)
2. for all \( a,b \in A \), \( 1 + axb \) is invertible.
3. for all \( a \in A \), \( 1 + ax \) is left-invertible.
4. for all \( a \in A \), \( 1 + xa \) is right-invertible.