By Denis S. Arnon, Bruno Buchberger
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Extra info for Algorithms in Real Algebraic Geometry
If we get ai = p from this for trivial reasons. 1. Namely, for ideals a1 , . . , an and a prime ideal p of R, we have the equivalence n n aj ⊂ p ⇐⇒ V j=1 aj ⊃ V (p). j=1 Indeed, the implication “ =⇒ ” is obtained by looking at the zeros of the ideals involved, whereas the implication “ ⇐= ” uses the formation of ideals I(·) of vanishing functions, as introduced above, in conjunction with rad(p) = p. Furthermore, we can use the assertion of Lemma 8 to show n n aj = V j=1 V (aj ). j=1 Thus, given an inclusion V (p) ⊂ nj=1 V (aj ), Lemma 8 yields the existence of an index i, 1 ≤ i ≤ n, such that V (p) ⊂ V (ai ).
An R-algebra (associative, commutative, and with a unit 1) is a ring A equipped with a structure of an R-module such that the compatibility rule r · (x · y) = (r · x) · y = x · (r · y) holds for all r ∈ R and x, y ∈ A where “ · ” denotes both, the ring multiplication and the scalar multiplication on A. ✲ B is a map that is a homomorphism A morphism of R-algebras A with respect to the ring and the module structures on A and B. ✲ A, we can easily view A as an Given any ring homomorphism f : R R-algebra via f ; just set r · x = f (r) · x for r ∈ R and x ∈ A.
Then N = g(g −1 (N )), as the image of a ﬁnitely generated R-module, must be ﬁnitely generated itself. Next, let M and M be Noetherian and let N ⊂ M be a submodule. Then N gives rise to the short exact sequence 0 ✲ f −1 (N ) ✲ N ✲ g(N ) ✲ 0 and it follows that f −1 (N ) ⊂ M as well as g(N ) ⊂ M are submodules of ﬁnite type. Applying Proposition 5 we see that N is of ﬁnite type. As a direct consequence we can show: Corollary 11. Let M1 , M2 be Noetherian R-modules. Then: (i) M1 ⊕ M2 is Noetherian.
Algorithms in Real Algebraic Geometry by Denis S. Arnon, Bruno Buchberger