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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