Discussion
Let \(\mathbb{R}[X^{2}, X^{3}]\) be the subring of \(\mathbb{R}[X]\) generated by \(X^{2}\) and \(X^{3}\). Consider the following statements.
I. The ring \(\mathbb{R}[X^{2}, X^{3}]\) is a unique factorization domain.
II. The ring \(\mathbb{R}[X^{2}, X^{3}]\) is a principal ideal domain.
Which one of the following is correct?
Answer: D
Both I and II are FALSE.
In \(R = \mathbb{R}[X^{2}, X^{3}]\) there is no element of degree 1, so \(X^{2}\) and \(X^{3}\) are irreducible (any factorisation would need a factor of degree 1). Then \(X^{6} = X^{2}\cdot X^{2}\cdot X^{2} = X^{3}\cdot X^{3}\) gives two factorisations into irreducibles with different numbers of factors, so \(R\) is not a UFD.
Every principal ideal domain is a unique factorization domain, so \(R\) is not a PID either. That rules out every option in which I or II is true.
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