Relationship between reduced rings, radical ideals and nilpotent elements

This post aims at providing some intuition and meaning for the following algebra relationship:

Reduced ring – Radical ideal – Nilpotent

Reduced ring – Radical ideal – Nilpotent

A basic fact of ring theory is that if you take a ring A and quotient it for a (double-sided) radical ideal I you get a reduced ring. Let us suppose A is a commutative ring and understand why this fact is true.

Nilpotent element
Def. a \in A is nilpotent \Leftrightarrow \exists n \in \mathbb{N} : a^n = 0

Informally, a nilpotent element is like a road ending in the middle of nowhere, collapsing in the depth of an abyss. You are driving on it, following the powers of a, and then all of a sudden, with no explanation, your road ends in a big black hole. Indeed, the zero really acts as some kind of black hole, attracting nilpotent-made roads at some point or another: we can think of nilpotent roads as spiraling into the zero.

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