Abstract Algebra II · Modules and finitely generated modules over a PID
Kernels and images are submodules
Problem
Let $f:M\to N$ be an $R$-module homomorphism. Prove directly that $\ker f$ is a submodule of $M$ and $\operatorname{im}f$ is a submodule of $N$.
Hint
If $f(x)=f(y)=0$, then $f(x-y)=0$.
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