# 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.