Question

Let \( {R}[X^2, X^3] \) be the subring of \( {R}[X] \) generated by \( X^2 \) and \( X^3 \). Consider the following statements: 1. The ring \( {R}[X^2, X^3] \) is a unique factorization domain.
2. The ring \( {R}[X^2, X^3] \) is a principal ideal domain.
Which one of the following is correct?

• A. Both I and II are TRUE
• B. I is TRUE and II is FALSE
• C. I is FALSE and II is TRUE
• D. Both I and II are FALSE

Hint:

For domain-related problems, verify integral closure for UFD and the ideal structure for PID.

Answer 1

The Correct Option is D

Step 1: Analyzing the unique factorization domain (UFD). The subring \( {R}[X^2, X^3] \) is not a unique factorization domain because it is not integrally closed. This property is essential for UFDs. Step 2: Analyzing the principal ideal domain (PID). The subring \( {R}[X^2, X^3] \) is not a principal ideal domain because it is not a free polynomial ring in one variable and hence does not satisfy the condition for every ideal to be principal. Step 3: Conclusion. Both statements are false. The correct answer is \( {(4)} \).