(1) Let $C_b(\mathbb{R})$ be the ring (with pointwise multiplication and addition) of bounded continuous functions.
Let $I_0=\{f_{(x)} \in C_b(\mathbb{R}) \space | \space lim_{x \to \pm \infty}f_{(x)} = 0 \}$ be an ideal.
Is $C_b(\mathbb{R})/I_0$ isomorphic to any other well known ring?
(2) Let $F(\mathbb{R})$ be the set of all absolutely integrable differentiable functions. $F(\mathbb{R})$ is an rng (without identity) under the operations of addition and convolution.
Let $I \subset F(\mathbb{R})$ be the subset of functions with bounded derivatives. For any $f \in I$ and $g \in F(\mathbb{R})$ the following holds:
$$(f*g)'=f'*g$$
$f'$ is bounded by definition of $I$ and $g$ is absolutely integrable which implies that $f'*g$ is bounded (It's pretty likely i got a mistake here somewhere). Thus $f*g \in I$ and convolution is commutative so $I$ is an ideal.
What does $F(\mathbb{R})/I$ look like?