homomorphism from a ring R to the quotient ring R/I

Question

Let $R$ be a commutative ring with $1$ and $I$ be an ideal. There is a natural homomorphism from $R$ to the quotient ring $R/I$ which maps $r$ to $r+I$. Is there any other homomorphism from $R$ to $R/I$?

Answer 1

Let $\pi: R \rightarrow R/I$ be the canonical ring homomorphism. If $R/I$ has a nontrivial endomorphism $\phi:R/I \rightarrow R/I$, then $\pi$ and $\phi\circ\pi$ are two distinct ring homomorphisms from $R$ to $R/I$. Indeed $\phi$ being nontrivial means that there is some $x+I$ with $\phi(x+I)\neq x+I$. Since $x+I$ is just another notation for $\pi(x)$ this shows our claim.

One example is given by $R=\Bbb C\times \Bbb C$, $I=\Bbb C \times \{0\}$ and $\phi:\Bbb C \rightarrow \Bbb C, x \mapsto \overline x$.