Question:
The ring of integers (Z, +, $\cdot$) is
The ring of integers (Z, +, $\cdot$) is
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$\mathbb{Z}$ is the prototype for an integral domain. It has everything a field has except multiplicative inverses for everyone.
Updated On: Jul 3, 2026
- an integral domain
- a field
- a skew field
- with zero divisors
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The Correct Option is A
Solution and Explanation
Step 1: Concept
An integral domain is a commutative ring with identity that has no zero divisors.
Step 2: Meaning
"No zero divisors" means if $a \cdot b = 0$, then either $a = 0$ or $b = 0$.
Step 3: Analysis
$(\mathbb{Z}, +, \cdot)$ is commutative and has identity 1. If the product of two integers is zero, at least one of them must be zero. It is not a field because non-zero elements like 2 do not have multiplicative inverses in $\mathbb{Z}$.
Step 4: Conclusion
Therefore, the ring of integers is an integral domain. Final Answer: (A)
An integral domain is a commutative ring with identity that has no zero divisors.
Step 2: Meaning
"No zero divisors" means if $a \cdot b = 0$, then either $a = 0$ or $b = 0$.
Step 3: Analysis
$(\mathbb{Z}, +, \cdot)$ is commutative and has identity 1. If the product of two integers is zero, at least one of them must be zero. It is not a field because non-zero elements like 2 do not have multiplicative inverses in $\mathbb{Z}$.
Step 4: Conclusion
Therefore, the ring of integers is an integral domain. Final Answer: (A)
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