Give \(\mathbb C^n\) coordinatewise addition and multiplication. Determine all surjective ring homomorphisms \[ \mathbb C^n\longrightarrow\mathbb C. \]
Solution
For \(1\leq j\leq n\), let \[ e_j=(0,\ldots,0,1,0,\ldots,0)\in\CC^n \] be the \(j\)th coordinate idempotent.
1. Any surjective ring homomorphism\[ \phi:\CC^n\to\CC \]satisfies\[ \phi(1,\ldots,1)=1. \]
Surjectivity gives \(x\in\CC^n\) with \(\phi(x)=1\). Then \[ \phi(1,\ldots,1) = \phi(1,\ldots,1)\phi(x) = \phi(x) =1. \] Thus the conclusion holds even under a convention in which a ring homomorphism is not assumed a priori to preserve the identity.
2. There is a unique index \(j\) such that\[ \phi(e_j)=1, \]while\[ \phi(e_k)=0 \]for every \(k\neq j\).
Each \(e_k\) is idempotent, so \(\phi(e_k)\) is an idempotent of the field \(\CC\). Hence \[ \phi(e_k)\in\{0,1\}. \] Moreover, \[ e_1+\cdots+e_n=(1,\ldots,1), \] so step 1 gives \[ \phi(e_1)+\cdots+\phi(e_n)=1. \] Thus at least one image equals \(1\). If \(j\neq k\), then \[ e_je_k=0, \] so \[ \phi(e_j)\phi(e_k)=0. \] Hence two distinct coordinate idempotents cannot both map to \(1\). Therefore the index is unique.
3. Fix the index \(j\) from step 2 and define\[ \iota_j:\CC\to\CC^n, \qquad \iota_j(z)=(0,\ldots,0,z,0,\ldots,0), \]and\[ \sigma\coloneqq\phi\circ\iota_j. \]Then for every \((z_1,\ldots,z_n)\in\CC^n\),\[ \phi(z_1,\ldots,z_n)=\sigma(z_j). \]
Write \[ (z_1,\ldots,z_n) = \sum_{k=1}^n\iota_k(z_k). \] If \(k\neq j\), then \[ \iota_k(z_k)=e_k\iota_k(z_k), \] so step 2 gives \[ \phi(\iota_k(z_k)) = \phi(e_k)\phi(\iota_k(z_k)) =0. \] Therefore only the \(j\)th summand survives: \[ \phi(z_1,\ldots,z_n) = \phi(\iota_j(z_j)) = \sigma(z_j). \]
4. The map \(\sigma:\CC\to\CC\) from step 3 is a fieldautomorphism.
2. Thus \(\sigma\) is a unital ring homomorphism from the field\(\CC\) to itself. Its kernel is therefore zero, so it is injective.
By step 3, the image of \(\phi\) equals the image of \(\sigma\). Since \(\phi\) is surjective, \(\sigma\) is surjective as well. Thus \(\sigma\) is a bijection preserving addition and multiplication, hence a field automorphism of \(\CC\).
5. Conversely, for every index \(j\) and every\(\sigma\in\Aut(\CC)\), the map\[ \phi_{\sigma,j}(z_1,\ldots,z_n) \coloneqq \sigma(z_j) \]is a surjective ring homomorphism \(\CC^n\to\CC\).
The coordinate projection \[ \pi_j:\CC^n\to\CC \] is a surjective ring homomorphism, and \(\sigma\) is a bijective ring homomorphism. Therefore \[ \phi_{\sigma,j} = \sigma\circ\pi_j \] is a surjective ring homomorphism.
6. Hence all surjective ring homomorphisms are exactly\[ \boxed{ (z_1,\ldots,z_n)\longmapsto\sigma(z_j), \qquad 1\leq j\leq n,\quad \sigma\in\Aut(\CC) }. \]
Steps <1>2–<1>4 show that every surjective ring homomorphism has the displayed form, and step 5 shows that every map of that form works. Such a map is \(\CC\)-linear exactly when \(\sigma=\operatorname{id}_{\CC}\), so the \(\CC\)-linear ones are the coordinate projections.
7. Q.E.D.
Step 6 is the complete classification.