If I may request the knowledge of the community on a point of trivia again, I was wondering if anyone could explain to me the etymology of reduced rings, that is, rings with no non-zero nilpotent elements.
In what sense is it actually reduced? In what sense has a process of reduction from an earlier state of things taken place?
Just a guess:
Given an algebraically closed field $k$, and an ideal $I\subseteq R = k[X_1, X_2, \ldots, X_n]$, the correspondences between the quotient ring $R/I$ and the set of points in $k^n$ where all elements of $I$ evaluate to $0$ is heavily studied, and it's one of the main backdrops to the entire mathematical field of algebraic geometry. These correpondences are so strong that from time to time terminology leaks through from one to the other.
I suspect that that has happened here. Nilpotent elements of $R/I$ corresponds to "multiple roots" (in a certain generalized sense) of $I$ in $k^n$. If $R/I$ is reduced, then the zero set of $I$ in $k^n$ is similarily reduced to only single roots.
The actual reduction operation we are talking about here would be to divide out by the nilradical, i.e. the ideal consisting of all the nilpotent elements.