examples of ring homomorphism $f : R → S$, where $R$ is arbitrary

Question

I've read the definition of the ring homomorphism:
Definition. Let $R$ and $S$ be rings. A ring homomorphism is a function $f : R → S$ such that:
(a) For all $x, y ∈ R, f(x + y) = f(x) + f(y).$
(b) For all $x, y ∈ R, f(xy) = f(x)f(y)$.
(c) $f(1) = 1.$

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I want to see some examples of ring homomorphism $f : R → S$, where $R$ is arbitrary. So, I consider $S=R/I$ , where $I$ is an ideal of $R$. I can also imbed $R$ in a Cartesian product.

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What else? Are there some famous ring homomorphisms $f : R \to S$ for $R$ arbitrary?

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Thank you.

Answer 1

$f : R \to R[x]$, where $x$ is an indeterminate.
$f : R \to S^{-1} R$, where $S$ is a multiplicatively closed subset of $R$ (localization).