Reduced ring

A reduced ring is a commutative ring $R$ such that the only nilpotent element of $R$ is $0$.

Equivalently, $R$ is reduced iff its nilradical is $0$. Reducedness is a basic “no infinitesimals” condition in algebraic geometry and commutative algebra.

Examples:

• $\mathbb Z$ and $k[x_1,\dots,x_n]$ are reduced.
• $k[x]/(x^2)$ is not reduced (the class of $x$ is nilpotent).
• Any finite product of reduced rings is reduced.