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, $RX\cap RY = R(XY)$.


If ideal with identity In this case you can find easily genrators of an ideals because ideal.equl to ring by using method of findng of genrator of ring . = {ra : r belongs to R } whare R is commutaitive ring with identity enter image description here if uo need explnation you can verify this by taking any iadeal with identity if you not understand you can ask about somw eaxmpls agian


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