# 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.

• A reduced ring is semiprime.

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

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

• 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 ReducedRing 2013-03-22 14:18:12 2013-03-22 14:18:12 CWoo (3771) CWoo (3771) 17 CWoo (3771) Definition msc 16N60 nilpotent-free