Jacobson Radical and Finite Dimensional Algebra In general, it is usually not the case that for a ring $R$, the Jacobson radical of $R$ has to be equal to the intersection of the maximal ideals of $R$.
However, what I do like to know is, if we are given a finite-dimensional algebra $A$ over the field $F$, is it true that the Jacobson radical of $A$ is precisely the intersection of the maximal ideals of $A$?

Yes, it is true. See
J.A. Drozd, V.V. Kirichenko, 
Finite dimensional algebras, 
Springer, 1994 
