### Compute radical and idempotents of a quotient algebra

I tried the following:

~~R.<x,y>=PolynomialRing(QQ,2)
I=Ideal(x^2,y^2)
S=R.quotient(I)
~~R.<x, y> = PolynomialRing(QQ, 2)
I = Ideal(x^2, y^2)
S = R.quotient(I)

I have the following question:

**I would like to compute with **~~SAGE ~~Sage the Jacobson radical of the algebra ~~S, ~~S,
all primitive orthogonal idempotents and the central idempotents.

~~ ~~

Of course, you can compute this by hand, but I am interested in ~~more ~~more
complicated examples, too (also in matrix algebras), but wanted ~~to ~~to
start with this simple example.

Since I am relatively new to Sage, I unfortunately do not know how to compute this.

I would be grateful for any help.