Let $F$ be field and $A=F[t]\setminus (t^2)$, where $(t^2)$ is the ideal of $F[t]$
(a) Show that every ideal of $A$ is principal ideal
(b) Find all prime ideals of $A$
I know $A$ is not integer domain because $t^2$ is reducible, So it is just commutative ring with unity. Thus, it shows that there exists non-integer domain which every ideal is a principal ideal.
To prove it, let $I$ be an ideal of $A$, I need to find one generator of $I$. But I couldn't. I don't think there is special theorem to solve it. I guess I just need to use the definition of ideal and the structure of the factor ring. Could anyone help me to solve it..? I just need a few hints. Thanks!