Radicals in primitive axial algebras

The paper contributes to the structure theory of primitive axial algebras. For a primitive axial algebra $A$ with a Frobenius form we compare the largest ideal $R(A)$ not containing any of the generating axes, the radical $A^{\perp }$ of the form, and the Jacobson radical $J(A)$, which we define simply as the intersection of all maximal ideals of $A$.