Find all the ideals and prime ideals of $F[x]times F[x]$ where $F$ is algebraically closed.

Question

Find all the ideals and prime ideals of $F[x]times F[x]$ where $F$ is algebraically closed.
So I am not sure what answer is expected in this question. $F[x]$ is a PID and so all of the ideals are of the form $(f(x))$ where $f$ is a polynomial. Thus all the ideals of $F[x]times F[x]$ are of the form $(f(x))times (g(x))$ where $g$. I do not think this can be simplified further. Next, IF $Itimes J$ is a prime ideal of $F[x]times F[x]$ then $I,J$ are prime ideals of $F[x]$. Prime ideals of $F[x]$ are maximal and are generated by irreducible polynomials. In $F$ these are the linear polynomials but $F[x]/(x-a)=F$ and $F times F$ is not a field. Thus it has no prime ideals

Tsemo Aristide · Answer

Let $p_i, i=1,2:F[x]times F[x]rightarrow F[x]$ the projection on the first and second factor. If $I$ is an ideal $p_i$ is an ideal. If $J$ is an ideal of $F[x], p_i^{-1}(J)$ is an ideal. If $p_1(I)$ is not $F[x]$ it is contained in $(x-a)$ and $p^{-1}(x-a)=(x-a)times F[x]$, same thing for $p_2$,
Remark that if $I$ is maximal, it contains $F[x]times 0$ or $0times F[x]$ otherwise since $(a,0)(0,b)=0$, this implies that maximal ideals are $(x-a)times F[x]$ or $F[x]times (x-a)$.

Find all the ideals and prime ideals of $F[x]times F[x]$ where $F$ is algebraically closed.

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