stylistic:style_transfer
Dedekind's theorem

Dedekind's theorem on the linear independence of characters. Let M be a monoid and k an integral domain, viewed as a monoid by considering the multiplication on k. Then any finite family (f) of distinct monoid homomorphisms f : M → k is linearly independent. In other words, every family (α) of elements α ∈ k satisfying

must be equal to the family (0).

Proof. First assume that k is a field, otherwise, replace the integral domain k by its quotient field, and nothing will change. We can linearly extend the monoid homomorphisms f : M → k to k - algebra homomorphisms F : k → k, where k is the monoid ring of M over k. Then, by linearity, the condition

yields

Next, for i, j ∈ I; i ≠ j the two k -linear maps F : k → k and F : k → k are not proportional to each other. Otherwise f and f would also be proportional, and thus equal since as monoid homomorphisms they satisfy: f (1) = 1 = f (1), which contradicts the assumption that they are distinct.

Therefore, the kernels Ker F and Ker F are distinct. Since k /Ker F ≅ F (k) = k is a field, Ker F is a maximal ideal of k for every i in I. Because they are distinct and maximal the ideals Ker F and Ker F are coprime whenever i ≠ j. The Chinese Remainder Theorem (for general rings) yields an isomorphism:

where

Consequently, the map

is surjective. Under the isomorphisms k /Ker F → F (k) = k, the map Φ corresponds to:

Now,

yields

for every vector (u) in the image of the map ψ. Since ψ is surjective, this means that

for every vector

Consequently, (α) = (0). QED.