Let $R$ be a ring, let $M$ and $N$ be left $R$-modules with zero elements $0_M \in M$ and $0_N \in N$, respectively, and let $f: M \to N$ be an $R$-module homomorphism. Then $f$ is injective if and only if
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\begin{align*}
\ker f = \{0_M\}.
\end{align*}
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Here $\ker f := \{m \in M : f(m)=0_N\}$, and $\{0_M\}$ denotes the zero submodule of $M$.