homomorphisms from a (semi)local ring to a ring with infinitely many maximal ideals

Question

In continuation of the this question: homomorphism from a (semi)local ring to $\mathbb Z$.

I tried to construct (unital) homomorphisms from a (semi)local ring to a ring with infinitely many maximal ideals. The obvious one is $f:k \to k [x_1,\ldots ,x_n, \ldots]$.
What about other cases?

Answer 1

A simpler one: $f:k\to k[x]$.

A more complicated one: $f:R\to R + xQ[x]$, where $R$ is a valuation domain and $Q$ is its field of fractions.

Answer 2

• The trivial example $0 \to R$ where $R$ is not semilocal.

• $\mathbb{F}_2 \to \mathbb{F}_4[X]$

  In general, the inclusion of a field $F$ in $E[X]$ where $E$ is a field extension of $F$.

• $\mathbb{Z}_{(2)} \to \mathbb{Z}_{(2)}[X]$ (the latter has infinitely many maximal ideals of the form $(2, p)$ where $p$ is a monic irreducible integer polynomial).

  In general, the inclusion of a semilocalization of $\mathbb{Z}$ (localization at the complement of a union of primes) in the corresponding polynomial ring.

• The diagonal homomorphism $\mathbb{F}_2 \to \prod_{i = 0}^\infty \mathbb{F}_2$ (defined by $x \mapsto (x, x, x, \cdots)$).

  This is a special case of the following classes of examples: take a semilocal ring $R$, an infinite index set $I$ and the diagonal map $R \to \prod_{i \in I} R$.

• The map $\mathbb{R} \to \mathcal{C}(\mathbb{R})$ (continuous real functions with pointwise operations) sending a scalar to the constant function.

• The above example and also other examples mentioned so far can be generalized as non-semilocal rings $R$ that contain a field $k$ as a subring, and taking the inclusion $k \hookrightarrow R$.

• Or, more generally, semilocal subrings of non-semilocal rings, such as $k \times k \hookrightarrow k[x] \times k[x]$.

Note that no example will be surjective, as a quotient of a semilocal ring is semilocal. Since non-surjective morphisms factor through a non-surjective injective map (from the quotient with the kernel to the original codomain), we can focus specifically on strict inclusions of subrings. So the last case mentioned above essentially accounts for all examples.