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A morphism of schemes into an affine scheme is uniquely determined by its homomorphism on global sections. Conversely, a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$ restricts on every affine open subscheme of $X$ to a map into $\mathrm{Spec}A$; equality on affine covers of overlaps glues these maps. No quasi-compactness or affineness assumption on $X$ is needed.

For a field $k$, a homomorphism of $k$-algebras is a unital ring homomorphism preserving scalar multiplication by every element of $k$. For unital algebras it is equivalently a ring homomorphism commuting with their structural maps from $k$. Pullback by a morphism of algebraic varieties gives such a homomorphism between their coordinate rings.

An algebra over a commutative ring $R$ is a ring $A$ with a specified unital ring homomorphism $R\to A$ whose image commutes with every element of $A$. Scalar multiplication is $r\cdot a=\phi (r)a$. For commutative $A$, the commutation requirement is automatic.

The category of unital rings has identity-preserving ring homomorphisms as morphisms. The zero ring, allowing $0=1$, is terminal, and $\mathbb{Z}$ is initial. Nonzero rings admit no maps from the zero ring. Monomorphisms are injective: maps from $\mathbb{Z}[x]$ probe individual elements. It is a regular category, with surjective homomorphisms as regular epimorphisms; these are distinct from arbitrary categorical epimorphisms.

Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.

For a ring homomorphism $R\to A$ and a left $R$-module $M$, the left $A$-action on ${\mathrm{Hom}}_R(A,M)$ is $(a\cdot h)(b)=h(ba)$. This construction is right adjoint to restriction of scalars.

A ring homomorphism transports ideals forward by extension and backward by inverse image, or contraction.

For a ring homomorphism $R\to A$, tensoring an $R$-module with the $(A,R)$-bimodule $A$ gives an $A$-module. This functor is left adjoint to restriction of scalars.

A field automorphism is a bijective ring homomorphism from a field to itself. The automorphisms fixing a specified subfield form a Galois group when the extension is Galois.

On an integral scheme $X$, every nonempty open $U$ contains its generic point $\eta$. Taking a germ gives an injective ring homomorphism from its regular functions to the function field ${\mathcal{O}}_{X,\eta }$. Injectivity follows on each nonempty affine open subscheme from the injection of an integral domain into its field of fractions, and then from the sheaf gluing axiom. This lets regular functions on different open subsets be compared inside one field.

For an injective ring homomorphism between nonzero integral domains $A\to B$, the scheme-theoretic fibre over the generic point of $\mathrm{Spec}A$ is $\mathrm{Spec}(S^{-1}B)$ with $S=A\setminus \{0\}$. Localization at these nonzero elements is a nonzero integral domain, so the generic fibre is a nonempty integral scheme. A surjective morphism between these affine schemes necessarily gives an injective ring map: a prime above $(0)$ contains its kernel.

The kernel of a ring homomorphism $\phi :R\to S$ is the ideal $\{r\in R:\phi (r)=0\}$. The ring homomorphism is injective exactly when this ideal is zero.