# Give an example of a commutative ring without zero-divisors that is not an integral domain.

Give an example of a commutative ring without zero-divisors that is not an integral domain.
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curwyrm
Integral domain:
Let D be a ring. Then D is an integral domain, provided these conditions hold:
1. D is a commutative ring.
2. D has a unity e, $e\ne 0.$
3. D has no zero divisors.
Let $2\mathbb{Z}=\left\{\dots ,-4,-2,0,2,4,\dots \right\}$, the set of all even integers, be the infinite commutative ring with usual addition and scalar multiplication.
By the definition of integral domain,
As $2\mathbb{Z}=\left\{\dots ,-4,-2,0,2,4,\dots \right\}$, there is no element $e\ne 0$, such that a*e=e*a=a, for all $a\in 2\mathbb{Z}.$
Thus, the ring $2\mathbb{Z}$ has no unity element.
Also, for $a\ne 0\in 2\mathbb{Z},$ a*0=0.
That is, for all $a\ne 0\in 2\mathbb{Z}$, there exists $b=0\in 2\mathbb{Z}$ such that a*b=0.
Thus, the infinite commutative ring $2\mathbb{Z}$ has no zero divisors.
As $2\mathbb{Z}$ has no unity element, $2\mathbb{Z}$ is not an integral domain.
Thus, $2\mathbb{Z}=\left\{\dots ,-4,-2,0,2,4,\dots \right\}$ is an infinite commutative ring, with no zero divisors, that is not an integral domain.
###### Not exactly what you’re looking for?
Jeffrey Jordon

Step 1

The example is $2\mathbb{Z}$ ,since it does not contain unity

Step 2

Notice that $2\mathbb{Z}$ is a subring of $\mathbb{Z}$ ,and hence it does not contain any zero-divisors