Let $\phi : R \to R'$ be a ring homomorphism with $S$ a subring of $R$. Let $I, J$ be ideals of $R$ with $I \subseteq J \subseteq R$.
- There is a one-to-one correspondence between the ideals of $R$ containing $\ker \phi$ and the ideals of $\phi(R)$.
- $R/\ker \phi \cong \phi(R)$, $(S+I)/I \cong S/(S \cap I)$, $\frac{R/I}{J/I} \cong R/J$, and $R/I \cong \phi(R)/\phi(I)$.
- The image of an ideal is not necessarily an ideal, but it will be if $\phi$ is surjective.
- $R/I$ has exactly two ideals iff $I$ is a maximal ideal.
Similar statements are true when $R, R'$ are groups, $\phi$ is a group homomorphism, $S$ is a subgroup of $R$, and $I, J$ are normal subgroups of $R$.
What are key distinctions between ideals and normal subgroups as mathematical objects in ring theory and group theory? Of course, ideals have a multiplication structure, so there are facts like $I = (1)$ iff $I$ contains a unit. But I'm hoping to learn about properties that are not an obvious consequence of the additional structure in ideals that is missing in normal subgroups.
For example, I don't think there is an analogy of the Jordan-Holder theorem for ideals. Additionally, a proper ideal is always contained in a maximal ideal by Zorn's lemma, but I'm not sure that a proper normal subgroup is always contained in a maximal normal subgroup (see this). Also normal subgroups correspond to normal extensions in Galois theory. And different operations on ideals have a geometric interpretation as different operations on subvarieties. One last distinction: if $G$ is a finite group and $N$ is a minimal normal subgroup of $G$ -- that is, $N$ is a nontrivial normal subgroup of $G$ that does not contain a (proper) nontrivial normal subgroup of $G$ -- then there is a finite simple group $L$ and $k \in \mathbb{Z}_{>0}$ such that $N \cong L^k = L \times L \times \dots \times L$.
Related: Reference request - ideals and normal subgroups in category theory.
