Maximal ideals

[Papapetros Vaggelis]

Let \(\displaystyle{\left(R,+,\cdot\right)}\) be a commutative ring with unity \(\displaystyle{1_{R}}\) .

Prove that, if \(\displaystyle{m}\) is a maximal ideal of this ring, then there exist a field \(\displaystyle{\mathbb{F}}\)

and an epimorphism \(\displaystyle{R\to \mathbb{F}}\) .

Conversely, if \(\displaystyle{R\to \mathbb{F}}\) is an epimorphism, where \(\displaystyle{\mathbb{F}}\)

is a field, then, find a maximal ideal of \(\displaystyle{\left(R,+,\cdot\right)}\) .

[Tsakanikas Nickos]

• Let \( \mathfrak{m} \) be a maximal ideal of \( R \). Then \( R/ \mathfrak{m} \) is a field and the canonical projection \( \pi \ \colon R \longrightarrow R / \mathfrak{m} \ , \ r \mapsto r + \mathfrak{m} \) is a ring epimorphism.
• Let \( f \ \colon R \longrightarrow \mathbb{F} \) be a ring epimorphism, where \( \mathbb{F} \) is a field. By the first isomorphism theorem, we have that \( R / \mathrm{Ker}(f) \cong \mathrm{Im}(f) = \mathbb{F} \), since \( f \) is an epimorphism. Therefore \( \mathrm{Ker}(f) \) is a maximal ideal of \( R \).