Example 2

Let K be a commutative field and E the algebra $K[[X_i]]_{i\in I}$ of formal power series in the variables $X_i$ with coefficients in K. By the previous example and Example 4 of VIII, p. 26, the radical of E consists of the formal power series with constant term zero. Observe that the ring E is an integral domain and that its radical is not reduced to 0, even though E is a subring of its field of fractions, which is without radical.