Ideals of rings of continuous functions with compact support.

Question

Here is the question that I am trying to solve:

Let $R$ be the ring of continuous functions from $\mathbb R$ to $\mathbb R.$ Let $I \subset R$ be the ideal of functions with compact support(i.e., $f(x) = 0$ for $|x|$ sufficiently large).\ $(a)$ Show that $I$ is not a prime ideal of $R.$\ $(b)$ Is there an ideal $J$ of $R$ that contains $I$ such that $J$ is prime? Why or why not?

My thoughts:

For part$(a).$

I know that this ring does not have an identity but how this will help me in showing that this $I$ is not prime, could someone explain this to me please or at least show me the proof.

For part$(b).$

I do not know how to even think about it, any help will be greatly appreciated.

Answer 1

For b: by Zorn's lemma, there exists a maximal ideal containing $I$. In particular, this maximal ideal is prime.