Which ring homomorphisms preserve/reflect what?

Question

Exams are coming up and I'm getting kind of desperate. So more now than ever, whatever help you're able to provide is much appreciated. In the abstract algebra exam I'm currently preparing for, there's a lot of focus on the following ring-theoretic concepts.

1. units
2. irreducible elements
3. prime elements
4. subrings
5. ideals
6. prime ideals
7. maximal ideals

Clearly these are all preserved and reflected under ring isomorphisms. But I need to know: which of these concepts are preserved and/or reflected under possibly injective and/or surjective ring homomorphisms? I'd work it out myself, but I really, really need to go on to the next section of the course (modules) in my revision.

Thank you for your time.

All answers are welcome, no matter how incomplete.

Answer 1

Here's what I got so far (rings are assumed to be commutative with identity).

Units are preserved by all ring homomorphisms, but not necessarily reflected by monomorphisms (take $\mathbb{Z}\to \mathbb{Q} $, e.g. $2\in \mathbb{Q}$ is not a unit in $\mathbb{Z}$) or epimorphisms (quotient $k[x] \to k$, $k$ a field).

Prime elements are not necessarily preserved under monomorphisms, take $\mathbb{Z} \to \mathbb{Z}[i]$, or $\mathbb{Z} \to \mathbb{Q}$.

The same example shows irreducible elements are not necessarily preserved under monomorphisms.

Subrings are preserved by all homomorphisms.

The image of an ideal under a monomorphism need not be an ideal (take $\mathbb{Z} \to \mathbb{Q}$, only the $(0)$ ideal gets mapped to an ideal).

The image of an ideal under an epimorphism is ideal.

The inverse image of an ideal is an ideal.

The inverse image of a prime ideal is prime.

The inverse image of a maximal ideal need not be maximal (take $\mathbb{Z} \to \mathbb{Q}$, $(0)\subset \mathbb{Q}$ is maximal).

Answer 2

dc2814's answer is almost complete. Let me just add that the counterexamples don't occur in special situations.

1. Prime elements are preserved by unramified homomorphisms.

2. The image of an ideal under a surjective ring homomorphism is an ideal. In fact, we have the following important correspondence principle: If $R \to S$ is a surjective homomorphism of rings, with kernel $K$, there is a 1:1 correspondence between ideals of $S$ and those ideals of $R$ which contain $K$.

3. If $R \to S$ is a homomorphism of finitely generated $k$-algebras (where $k$ is a field), then it pulls back maximal ideals to maximal ideals. This follows from Zariski's Lemma.