A reduced finitely generated $k$-algebra

A reduced finite-type $k$-algebra is a commutative unital $k$-algebra $A$ admitting a finite list $a_1,\dots ,a_n$ such that every element is a polynomial in that list with coefficients from $k$, and such that $a^m=0$, $m\ge 1$, implies $a=0$. With the terminology of The nilradical and reduced rings and Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, equivalently finite type means admitting a surjection $k[T_1,\dots ,T_n]\to A$. The zero algebra is included. “Module-finite” instead means finitely generated as a $k$-module; finite type does not impose that condition.

Source comparison: Milne, Algebraic Geometry, v6.10, §3e pp. 65–66. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.