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For a morphism of ringed spaces, the pullback of a sheaf of modules is left adjoint to the direct image sheaf functor. The unit sends a local section $h$ to $1\otimes h$ after inverse image. On the structure sheaf it becomes the given ring homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.

Coefficientwise reduction defines a ring homomorphismIt preserves sums, products, and divisibility. The Frobenius endomorphism gives $\overline{g(X^p)}=\bar{g}(X{)}^p$.

For a ring homomorphism $\theta :R\to A$, an $A$-module becomes an $R$-module by the displayed action. Its underlying abelian group stays the same.

A bijective ring endomorphism. Its inverse is again a ring homomorphism, so it preserves the ring structure in both directions.

A ring homomorphism from a ring to itself. Surjectivity need not imply injectivity in general, but a surjective ring endomorphism of a Noetherian ring is a ring automorphism.

A ring isomorphism is a bijective ring homomorphism. It preserves all ring operations and has a ring-homomorphic inverse.