## Nilpotent Ideal and Surjective Module Homomorphisms

## Problem 431

Let $R$ be a commutative ring and let $I$ be a nilpotent ideal of $R$.

Let $M$ and $N$ be $R$-modules and let $\phi:M\to N$ be an $R$-module homomorphism.

Prove that if the induced homomorphism $\bar{\phi}: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.

Add to solve later