Question

Let R be an integral domain. Which of the following is true?

Question 7Answer

a.
R is an empty set


b.
R is finite


c.
R is a field


d.
R has no zero divisors

Answer 1

To determine which of the statements is true for an integral domain \( R \), let's review what an integral domain is. An integral domain is a commutative ring with unity (1 ≠ 0) that has no zero divisors. This means that if \( a \) and \( b \) are non-zero elements in \( R \), then their product \( ab \) is also non-zero.

Now let's evaluate each option:

a. **R is an empty set**
This is false. An integral domain must have at least one element (the multiplicative identity, which is 1).

b. **R is finite**
This is false. An integral domain can be either finite or infinite. For example, the set of integers \( \mathbb{Z} \) is an integral domain and it is infinite.

c. **R is a field**
This is false. While every field is an integral domain, not every integral domain is a field. An integral domain needs not have every non-zero element invertible.

d. **R has no zero divisors**
This is true. This is actually one of the defining properties of an integral domain.

Based on this analysis, the correct answer is:

**d. R has no zero divisors**.