Regarding the ideals of the ring $R^{\infty}$

Question

$\mathbf{Question:}$ For every ring $R$, define $R^{\infty}$ as the infinite sequences of the elements of $R$ with pointwise multiplication and addition. Which of the following is correct?

1. For any ring $R$, every ideal of $R^{\infty}$ can be expressed as $(t)$ for some $t\in R^{\infty}$
2. The first option is incorrect, but every ideal of $\mathbb{Z}^{\infty}$ can be expressed as $(t)$ for some $t\in\mathbb{Z}^{\infty}$.
3. For any ring $R$ every ideal of $R^{\infty}$ can be expressed as $(t_0,t_1,\dots, t_k) $ for $t_i \in R^{\infty}$.
4. None fo the above.

$\mathbf{Attempt:}$ For the first and second choice, assume $R=\mathbb{Z}$ Consider $(4,8,0,0,...)=(2,4,0,0,...)((2,2,0,0,...)=(2,4,1,0,...)((2,2,0,0,...)$ in $R^{\infty}$. So, it's not a UFD $\implies$ $R^{\infty}$ is not a PID. [$(2,4,0,0,...)$ and $(2,4,1,0,...)$ are not associates]. Therefore both are false.

Now consider $R=\mathbb{Z}$ (in regard to option 3). Consider the ascending chain of ideals $I_n=\mathbb{Z} \times \underbrace{...}_\text{n times}\times \mathbb{Z} \times 0 \times 0 ...$ in $R^{\infty}$ which does not terminate. Hence, $R^{\infty}$ is not Noetherian $\implies$ not every ideal of $R^{\infty}$ is finitely generated.

So option $4$ is the correct choice.

Is this correct? (Verification & Correction Request)