Past exam of the commutative algebra course of the University of Oxford / 2023 / 3 / a / Solution

A ring is reduced if its only nilpotent element is zero. Equivalently,The second expression says that its nilradical vanishes. A reduced ring may have zero divisors; for example, the direct product of rings $\mathbb{R}\times \mathbb{R}$ is reduced but $(1,0)(0,1)=0$.