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

emancipezN 2020-11-29 Answered
Give an example of a commutative ring without zero-divisors that is not an integral domain.
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curwyrm
Answered 2020-11-30 Author has 87 answers
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, e0.
3. D has no zero divisors.
Let 2Z={,4,2,0,2,4,}, the set of all even integers, be the infinite commutative ring with usual addition and scalar multiplication.
By the definition of integral domain,
As 2Z={,4,2,0,2,4,}, there is no element e0, such that a*e=e*a=a, for all a2Z.
Thus, the ring 2Z has no unity element.
Also, for a02Z, a*0=0.
That is, for all a02Z, there exists b=02Z such that a*b=0.
Thus, the infinite commutative ring 2Z has no zero divisors.
As 2Z has no unity element, 2Z is not an integral domain.
Thus, 2Z={,4,2,0,2,4,} is an infinite commutative ring, with no zero divisors, that is not an integral domain.
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Jeffrey Jordon
Answered 2021-09-28 Author has 2064 answers

Step 1

The example is 2Z ,since it does not contain unity

Step 2

Notice that 2Z is a subring of Z ,and hence it does not contain any zero-divisors

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