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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.

A reduced ring is semiprime.

A commutative semiprime ring is reduced. In particular, all integral domains and Boolean rings are reduced.
An example of a reduced ring with zerodivisors 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
$\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}0&1\\ 0&0\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}.$ 
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clarify by Mathprof ✘
slight change by Mathprof ✘
concrete example. by Algeboy ✓
an additional part in third item by jocaps ✘