So say I have two rings, $R_1$ and $R_2$, and I have a homomorphism between them, $\phi$.
There are sources, such as my lecturer, who said in passing that the homomorphism $\phi$ can be factorized as a composition of a surjective map and an injective map$$R_1 \to \phi(R_1) \hookrightarrow R_2.$$The first arrow is $\phi$ and onto, of course. And the image $\phi(R_1)$ is a subring of $R_2$, while it is a quotient ring of $R_1$.
What is the significance of this, and how do I see this? It seems mysterious. I'll rattle off a few of my thoughts.
- The first arrow is onto, so everything in the image, i.e. $R_1$ gets hit. No one gets left out. And the cardinality of $R_1$, "morally", is at least as big as that of $\phi(R_1)$ if we want to preserve the structure. Indeed, $\phi(R_1)$ is a quotient ring of $R_1$, which should indicate it's like a miniature version of $R_1$.
- The second arrow is into, so $1$-to-$1$, no funny business like $y = x^2$, etc. It's sort of the opposite of the first map. "Morally" the cardinality of $R_2$ is at least as big as that of $\phi(R_1)$. And yeah, $\phi(R_1)$ is just the image living inside this ambient space of $R_2$. Hence subring.
I guess I am curious, what is the usefulness of this factorization for algebraic geometry (which a lot of people on this website talk about), i.e. what important problems does it solve there?