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 
and quotient it for a (double-sided) radical ideal 
you get a reduced ring. Let us suppose A is a commutative ring and understand why this fact is true.
Nilpotent element
Def. 
is nilpotent 
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 
, 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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