How to approach construction of ring homomorphisms? As a first year math undergrad I'm taking my first abstract algebra course and I've been struggling with one aspect of the course in particular.
How do you approach the construction of certain ring homomorphisms / isomorphisms, in general, given certain restrictions?
Given a function, it is easy for me to evaluate whether or not it is a homomorphism. However, when asked, for example:

GIVE EXAMPLES OF
-commutative rings R, S and a ring homomorphism f : R → S such that Kerf is a maximal ideal, and S is not a field.
-commutative rings R, S and a ring homomorphism f : R → S such that Kerf is not a
maximal ideal, and S is a field.

I never really know where to start. How do you "incorporate" different conditions to find homomorphisms? Is it just a matter of knowing a lot of examples, or is there a more solid approach to these problems?
Any and all help is appreciated!
 A: Try to break down the problem into pieces.  Some of it is luck and experience, but it is also trying to make the problem reasonable.


*

*In this case, you have a ring homomorphism $f:R\rightarrow S$ whose kernel is a maximal ideal of $R$ and $S$ is not a field.  So, what do you know; since the kernel of $f$ is a maximal ideal, the image of $f$ is a field.  Since $S$ is not a field, we know that $S$ can't be the image.  Hence, we have a field $F$ so that $f$ can be written as
$$
R\rightarrow F\hookrightarrow S.
$$
In other words, the image of $R$ is $F$ (and $F\simeq R/\ker(f)$) and $F\subseteq S$.  So, now we start thinking about rings that contain fields.  An easy example would be the polynomial ring $\mathbb{C}[X]$ because it contains the field $\mathbb{C}$ while it is not a field.  Now, we need a map whose image is $\mathbb{C}$.  Well, the identity map from $\mathbb{C}$ to itself will work (and automatically have a maximal ideal as a kernel because the image is a field).  This gives
$$
\mathbb{C}\xrightarrow{\sim}\mathbb{C}\hookrightarrow\mathbb{C}[x].
$$
A more sophisticated example would be the map
$$
\mathbb{R}[x]\rightarrow\mathbb{R}[x]/(x^2+1)\simeq\mathbb{C}\hookrightarrow\mathbb{C}[x].
$$

*On the other hand, taking apart the other statement, we have a map from $R$ to $S$ and the kernel of the map is not a maximal ideal.  Since the kernel is not a maximal ideal, the image is not a field.  So, you need a field that contains a ring which is not a field.  A nice example of this would be $\mathbb{Q}$ containing $\mathbb{Z}$.  Then, you could use the composition of maps
$$
\mathbb{Z}\xrightarrow{\simeq}\mathbb{Z}\hookrightarrow\mathbb{Q}.
$$
The kernel of these maps is the zero ideal, which is not maximal in $\mathbb{Z}$.
In each of these examples, the idea is to make the first map have the right kernel property and then make the second map place the image in the right type of codomain.  By making the second map injective, you preserve the properties of the kernel.
