A similar question like this has been asked here, apologies, but need to clarify something at the end
Our homework question was to show that any ring homomorphism $f:K\rightarrow R $ (where K is a field, R is a ring) is injective.
So by definition of a ring homomorphism, $ker(f)$ is an ideal in K.
By K a field $\Rightarrow$ the only ideals are $\{0\}, K$
$ker(f)={0} \Rightarrow f$ is injective, by definition of an injective ring homomorphism.
However I answered saying that $ker(f)=K \Rightarrow$ f is the zero map, $f(k)=0\ \forall k\in K$. So $f$ is either injective, or it is the zero map.
To which my corrector asked, " Is the zero map a ring homomorphism?"
Obviously if you've got something like $f:\Bbb R \rightarrow \Bbb Z_4 $, then $f(1_R)=f(1)=0 \ne 1 $ so this can't be a ring homomorphism
But if you considered $f:\Bbb R \rightarrow \{0\} $ , then $f(1_R)=f(1)=0 = 1_0 $
Does this work as a ring homomorphism?
So can the zero map sometimes be a ring homomorphism?
Best Answer
We have to distinguish between two algebraic structures with appropriate homomorphisms between them (in fact, they constitute categories)
$\bullet$ Rings (which I always assume to be unital) with homomorphisms of rings.
As always, a homomorphism of blupp preserves the whole structure of blupp, so in particular for blupp=Ring the unit is preserved by definition. This definition is (or should be ...) universally accepted.
$\bullet$ Rngs or non-unital rings with homomorphisms of rngs
Following the general principle, a homomorphism of rngs preserves the addition and the multiplication (and then also additive inverses and the zero), but not the unit because there is none.
There is a so-called forgetful functor from Rings to Rngs which forgets the unit. Notice that if $R$ is a ring, the underlying rng $|R|$ is a different object (and similarly for homomorphisms). This is usually ignored in textbooks and lectures, leading to utter confusions. Similar problems arise with other forgetful functors. These problems will end immediately when we take forgetful functors seriously. Unfortunately, some authors consider rings with unit but consider also non-unital homomorphisms between them, which doesn't make sense at all because this actually ignores the general principles of universal algebra, secretly applies the forgetful functor all the time, and doesn't take rings seriously. And some authors even don't consider $0$ as a ring because "it has no unit", sic! Of course the zero ring is a ring. It's quite cold these days, so books following this approach would make a warm fire ...
Anyway, now the question is easily answered:
Let $f : R \to R'$ be a homomorphism of rings (thus preserving the unit by definition). Then $f=0$ iff $R'=0$. It doesn't matter if $R$ is a field or not, the condition doesn't depend on $R$ at all.
On the other hand, there is always a zero homomorphism of the underlying rngs $0 : |R| \to |R'|$. Notice, again, that these are different objects than the rings themselves!