Algebra 819 - Homework 7
نویسنده
چکیده
Proof. Let F be the field of fractions of R. By 6.2.8, we must show that R = Int(R,F). The inclusion R ⊆ Int(R,F) is immediate. Let f/g ∈ Int(R,F) and let it be written in lowest terms. Suppose to the contrary that f/g / ∈ R. Then g = q1 · · · q` with qj prime for 1 ≤ j ≤ `. We can also write f = p1 · · · pk where the pi are primes for 1 ≤ i ≤ k. Note that pi 6∼ qj for any i, j because f/g is in lowest terms. Since f/g is integral over R, there exists a monic polynomial in R[x] with f/g as a root. That is, for some collection of coefficients in R we have ( f
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Algebra 819 - Homework 6
(a) Let ¦ : V × S → V, (v, s) → v ¦ s be a function. Define ¦op : Sop × V → V, (s, v) → v ¦ s. Then ¦ is an R-linear right action of S on V if and only if ¦op is an R-linear action of Sop on V . (⇒) Suppose ¦ is an R-linear right action of S on V . By definition, for all s, t∈ S, u, v∈ V and r∈ R we have (i) v ¦ (st) = (v ¦ s) ¦ t (ii) v ¦ e = v (iii) (u + v) ¦ s = u ¦ s + v ¦ s (iv) (rv) ¦ s =...
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