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}.