What does the kernel of a semiring homomorphism being a null set signify?

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According to wikipedia, the kernel of a homomorphism measures the degree to which a homomorphism fails to be injective.

What does it mean when a kernel of semiring is a null set? That is when $f:S\longrightarrow S'$ is a homomorphism between semirings $S$ and $S'$ such that $ker(f)=\lbrace \rbrace$, a null set. Is it true that a homomorphism whose kernel is a null set always a one-one mapping?