During my algebra lecture, my lecturer used the fact that any nontrivial commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that?
Thanks
2026-04-12 09:41:40.1775986900
Commutative ring can be homomorphically mapped onto field
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Every nontrivial commutative (unital) ring $R$ has a maximal ideal, $\mathcal m$, and the natural map $R\to R/\mathcal m$ is an onto map from $R$ to a field.
The existence of a maximal ideal for an arbitrary non-trivial commutative ring is equivalent to the Axiom of Choice.