New PDF release: A course in commutative algebra

By Ash R.B.

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Example text

By hypothesis, I is finitely generated, say by homogeneous elements a1 , . . , ar of degree n1 , . . , nr respectively. Let R = R0 [a1 , . . , ar ] be the R0 -subalgebra of R generated by the ai . It suffices to show that Rn ⊆ R for all n ≥ 0 (and therefore R = R ). We have R0 ⊆ R by definition of R , so assume as an induction hypothesis that Rd ⊆ R for d ≤ n − 1, where n > 0. If a ∈ Rn , then a can be expressed as c1 a1 + · · · + cr ar , where ci (i = 1, . . , r) must be a homogeneous element of degree n − ni < n = deg a.

Therefore bn = bn+1 t = ctbn for some c ∈ R, so (1 − ct)bn = 0. But t ∈ M, so t is not a unit, and consequently ct = 1. Thus bn must be 0, and we have a = bn tn = 0, proving the claim. Now let I be any nonzero proper ideal of R. Then I ⊆ M, but by the above claim n n n+1 . we have I ⊆ ∩∞ n=0 M . 3. DISCRETE VALUATION RINGS 9 Choose a ∈ I \ Mn+1 ; since Mn = (t)n = (tn ), we have a = utn with u ∈ / M (because a∈ / Mn+1 ). But then u is a unit, so tn = u−1 a ∈ I. To summarize, I ⊆ Mn = (tn ) ⊆ I, proving that I is principal.

This contradicts our choice of t (recall that t ∈ S \ Q2 ). ♣ Chapter 3 Valuation Rings The results of this chapter come into play when analyzing the behavior of a rational function defined in the neighborhood of a point on an algebraic curve. 4, we generalized a result about field extensions to rings. Here is another variation. 1 Theorem Let R be a subring of the field K, and h : R → C a ring homomorphism from R into an algebraically closed field C. If α is a nonzero element of K, then either h can be extended to a ring homomorphism h : R[α] → C, or h can be extended to a ring homomorphism h : R[α−1 ] → C.

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A course in commutative algebra by Ash R.B.


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