This is a question specific to the terminology used in Eisenbud and Harris's "The Geometry of Schemes". My question has to do with the meaning of the expression "affine variety" (over an algebraically closed field $K$) in that book, compared to say, the meaning of the same expression in Hartshorne's "Algebraic Geometry".
It appears to me that Hartshorne's affine varieties over $K$ are irreducible, while affine varieties over $K$ in the sense of Eisenbud-Harris satisfy a weaker condition (that the coordinate ring is a finitely generated reduced algebra over $K$).
Does that mean for instance that, over $\mathbb{C}$, in the affine plane with coordinates $x$ and $y$, if one considers $xy=0$, then this would define a complex affine variety in the sense of Eisenbud-Harris, but would not in the sense of Hartshorne?
I just would like a confirmation of what I suspect to be true, or a correction of my understanding regarding the meaning of "affine varieties".