reduced ring
A ring $R$ is said to be a reduced ring if $R$ contains no nonzero nilpotent elements^{}. In other words, ${r}^{2}=0$ implies $r=0$ for any $r\in R$.
Below are some examples of reduced rings.

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A reduced ring is semiprime^{}.

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A ring is a domain (http://planetmath.org/CancellationRing) iff it is prime (http://planetmath.org/PrimeRing) and reduced.

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A commutative^{} semiprime ring is reduced. In particular, all integral domains^{} and Boolean rings^{} are reduced.

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Assume that $R$ is commutative, and let $A$ be the set of all nilpotent elements. Then $A$ is an ideal of $R$, and that $R/A$ is reduced (for if ${(r+A)}^{2}=0$, then ${r}^{2}\in A$, so ${r}^{2}$, and consequently $r$, is nilpotent^{}, or $r\in A$).
An example of a reduced ring with zerodivisors is ${\mathbb{Z}}^{n}$, with multiplication^{} defined componentwise: $({a}_{1},\mathrm{\dots},{a}_{n})({b}_{1},\mathrm{\dots},{b}_{n}):=({a}_{1}{b}_{1},\mathrm{\dots},{a}_{n}{b}_{n})$. A ring of functions taking values in a reduced ring is also reduced.
Some prototypical examples of rings that are not reduced are ${\mathbb{Z}}_{8}$, since ${4}^{2}=0$, as well as any matrix ring over any ring; as illustrated by the instance below
$$\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$ 
Title  reduced ring 

Canonical name  ReducedRing 
Date of creation  20130322 14:18:12 
Last modified on  20130322 14:18:12 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  17 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16N60 
Synonym  nilpotentfree 