Does the Subring Generated by Two UFDs Always Result in a UFD?

[Petek]

Let $R$ be an integral domain. Suppose that $R_1$ and $R_2$ are proper subrings of $R$ and that both $R_1$ and $R_2$ are unique factorization domains (UFDs). Let $R_3$ be the subring of $R$ that is generated by $R_1$ and $R_2$. Is $R_3$ necessarily a UFD? (The subring generated by two subrings is defined to be the intersection of all subrings containing the two given subrings.)

 

[Petek]

My solution:

[sp]$\mathbb{Z}[X^2]$ and $\mathbb{Z}[X^3]$ are both UFDs. However, the subring they generate in $\mathbb{Z}[X]$, namely $\mathbb{Z}[X^2, X^3]$, is not a UFD because $X^6=X^2\cdot X^2\cdot X^2$ and $X^6=X^3 \cdot X^3$ are two distinct factorizations of $X^6$ into irreducible elements in $\mathbb{Z}[X^2, X^3]$.[/sp]