Ring homomorphism

A ring homomorphism preserves addition, multiplication, and, under the convention used here, the multiplicative identity.

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

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.

For a ring homomorphism $f:R\to S$, extension sends an ideal $I\subseteq R$ to the ideal $Sf(I)$ generated by its image. Contraction sends an ideal $J\subseteq S$ to $f^{-1}(J)$. These are distinct forward and inverse-image operations.

For a ring homomorphism $f:R\to A$ and an ideal $I\subseteq R$, the extension $I^e=IA$ is the ideal of $A$ generated by $f(I)$.

For a ring homomorphism $f:R\to A$ and an ideal $J\subseteq A$, its contraction is $J^c=f^{-1}(J)$.

For an element $a$ of a commutative $R$-algebra, evaluation on a polynomial ring is the ring homomorphism

Every ring homomorphism $\varphi :R\to S$ induces an isomorphism