Question

Which of the following is not an integral domain?

• A. Ring of Gaussian integers
• B. Ring of Integers
• C. Ring of real numbers
• D. Ring of all \(n\times n\) matrices whose elements are integers

Hint:

Matrix rings are generally not integral domains because matrix multiplication is not commutative and zero divisors may exist.

Answer 1

The Correct Option is D

Concept:
An integral domain is a commutative ring with unity and without zero divisors. A matrix ring is generally not an integral domain because matrix multiplication is not commutative, and non-zero matrices can multiply to give the zero matrix.

Step 1: Check ring of integers. The ring \[ \mathbb{Z} \] has no zero divisors. So it is an integral domain.

Step 2: Check ring of real numbers. The ring \[ \mathbb{R} \] is a field. Every field is an integral domain.

Step 3: Check Gaussian integers. The Gaussian integers \[ \mathbb{Z}[i] \] form an integral domain.

Step 4: Check matrix ring. The ring of all \(n\times n\) matrices over integers is not commutative for \(n\geq 2\). Also, it can have zero divisors. Therefore, it is not an integral domain.

Step 5: Final answer. \[ \boxed{\text{Ring of all }n\times n\text{ matrices whose elements are integers}} \]