By Gregor Kemper

This textbook deals an intensive, sleek advent into commutative algebra. it truly is intented regularly to function a advisor for a process one or semesters, or for self-study. The rigorously chosen subject material concentrates at the strategies and effects on the heart of the sphere. The e-book continues a continuing view at the typical geometric context, allowing the reader to achieve a deeper realizing of the cloth. even though it emphasizes idea, 3 chapters are dedicated to computational points. Many illustrative examples and workouts enhance the text.

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**Additional info for A Course in Commutative Algebra**

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Assume P ∈ / V(I), so there exists G. 1007/978-3-642-03545-6 4, c Springer-Verlag Berlin Heidelberg 2011 33 34 3 The Zariski Topology f ∈ I with f (P ) = 0. We need to show that P ∈ V(J), so let g ∈ J. Then f g ∈ I ∩ J, so f (P )g(P ) = 0. But this implies g(P ) = 0. Part (b) is clear. 1 tells us that ﬁnite unions and arbitrary intersections of aﬃne varieties in K n are again aﬃne varieties. Since K n and ∅ are also aﬃne varieties, this suggests that we can deﬁne a topology using the aﬃne varieties as closed sets.

M with Mi ⊆ Mi+1 for all positive integers i, there exists an integer n such that Mi = Mn for all i ≥ n. In other words, every strictly ascending chain of submodules is ﬁnite. (b) R is called Noetherian if R is Noetherian as a module over itself. In other words, R is Noetherian if the ideals of R satisfy the ascending chain condition. , for submodules M1 , M2 , M3 , . . ” G. 1007/978-3-642-03545-6 3, c Springer-Verlag Berlin Heidelberg 2011 23 24 2 Noetherian and Artinian Rings all positive integers i, there exists an integer n such that Mi = Mn for all i ≥ n.

However, Noetherian algebras are not always ﬁnitely generated. So graded rings constitute a special case in which this converse holds. 8 (The Noether property and subrings). 1 we have seen that in general the Noether property does not go down to subrings. In this exercise we look at a situation in which it does. (a) Let S be a Noetherian ring and R ⊆ S a subring such that there exists a homomorphism ϕ: S → R of R-modules with ϕ|R = idR . Show that R is Noetherian, too. (b) Show that for a ring R, the following three statements are equivalent: (i) R is Noetherian; (ii) R[x] is Noetherian; (iii) R[[x]] is Noetherian.