reduced ring

A ring $R$ is said to be a reduced ring if $R$ contains no non-zero 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 zero-divisors is $\mathbb{Z}^{n}$ , with multiplication defined componentwise: $(a_{1},\ldots,a_{n})(b_{1},\ldots,b_{n}):=(a_{1}b_{1},\ldots,a_{n}b_{n})$ . 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

$\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}0&1\\ 0&0\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}.$

Title	reduced ring
Canonical name	ReducedRing
Date of creation	2013-03-22 14:18:12
Last modified on	2013-03-22 14:18:12
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	17
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16N60
Synonym	nilpotent-free