# The Unapologetic Mathematician

## Ring homomorphisms

There is a special kind of function between rings, just like we have in groups. Given rings $R$ and $S$, a function $f:R\rightarrow S$ is called a homomorphism if it preserves all the ring structure.

The sort of odd thing here is that we’ve got two different kinds of rings to consider: those with and without identities. If we’re considering rings in general, we require that

• $f(r_1+r_2)=f(r_1)+f(r_2)$
• $f(r_1r_2)=f(r_1)f(r_2)$

but if we’re restricting ourselves to rings with identities, we also require that

• $f(1)=1$

where the $1$ on the left is the identity of $R$, and the one on the right is the identity for $S$. If we have two rings with identities but we consider them as general rings there will be more homomorphisms than if we consider them as rings with identity. It becomes important to pay a bit of attention to what kind of rings we’re really concerned with.

As an exercise, consider an arbitrary ring $R$ and see what ring homomorphisms exist from $\mathbb{Z}$ to $R$. If $R$ has an identity, which of these homomorphisms preserve the identity?

Oh, and I probably should mention this: all the terminology from groups comes along for the ride. An injective (one-to-one) ring homomorphism is a monomorphism. A surjective (onto) ring homomorphism is an epimorphism. One that’s both is an isomorphism. A homomorphism from a ring to itself is an endomorphism, and an isomorphism from a ring to itself is an automorphism.