# 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)
```

I have the following question:

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

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