# Finding a generator for a principal ideal

Let $$R:= \mathbb{Q}[X,Y]$$ be a ring of polynomials.

My question is whether $$RX \cap RY$$ and $$\{Xa+Yb \mid a,b \in R \}$$ are principal ideals by giving a generator.

Definition of the generator: Let $$R$$ be a ring. An ideal $$I$$ is a principal if there exists $$g \in I$$ so that $$I= Rg .$$

This $$g$$ is called generator.

I am not sure hot to find such g..also because I am confused about $$RX$$ and $$RY$$. How does the set $$\mathbb{Q}[X,Y] X$$ look like?

$$X,Y$$ are coprime in $$\Bbb Q[X,Y]$$. Therefore any generator of $$I:=\;$${$$Xa+Yb | a,b\in\Bbb Q[X,Y]$$} would be a common divisor of $$X,Y$$ and thus constant. Since $$I$$ is a proper ideal, i.e. $$f(0,0) = 0$$ for all $$f(x,y)\in I$$, then $$I$$ does not admit constants as they are units. Therefore, $$I$$ is not a principal ideal.
Finally, again because $$X,Y$$ are coprime and $$\Bbb Q[X,Y]$$ is a unique factorization domain then any polynomial divisible by both $$X,Y$$ is also divisible by $$XY$$ which itself is a member of the ideal $$RX\cap RY$$ made of the common multiples of $$X$$ and $$Y$$. Therefore, $$RX\cap RY = R(XY)$$.