A Course in Commutative Algebra by Gregor Kemper

By Gregor Kemper

This textbook deals an intensive, glossy creation into commutative algebra. it truly is intented mostly to function a advisor for a process one or semesters, or for self-study. The conscientiously chosen material concentrates at the options and effects on the heart of the sphere. The booklet keeps a relentless view at the traditional geometric context, allowing the reader to achieve a deeper realizing of the cloth. even though it emphasizes idea, 3 chapters are dedicated to computational features. Many illustrative examples and workouts enhance the text.

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Conversely, assume that I is a prime ideal. ,xn ] (∅) = K[x1 , . . , xn ]. To show that X is irreducible, let X = X1 ∪X2 with Xi closed in X, so Xi = X ∩VK n (Ii ) with Ii ⊆ K[x1 , . . , xn ] ideals. 1(a) for the equality. ,xn ] (X) = I. Since I is a prime ideal, there exists i with Ii ⊆ I, so X ⊆ VK n (I) ⊆ VK n (Ii ). This implies Xi = X. Therefore X is irreducible. (b) The proof of this part is obtained from the proof of part (a) by changing K[x1 , . . ” The following theorem allows us to view irreducible spaces as the “atoms” of a Noetherian space.

5 (Jacobson rings). Show that for verifying that a ring R is a Jacobson ring it is enough to check that every prime ideal P ∈ Spec(R) is an intersection of maximal ideals. 6 (Z is a Jacobson Ring). Show that the ring Z of integers is a Jacobson ring. 7 (Explicit computations with a variety). Consider the ideal I = x41 + x42 + 2x21 x22 − x21 − x22 ⊆ R[x1 , x2 ]. (a) Determine X := V(I) ⊆ R2 and draw a picture. (b) Is I a prime ideal? Is I a radical ideal? 17) hold for I? 8 (Colon ideals). If I and J ⊆ R are ideals in a ring, the colon ideal is defined as I : J := {a ∈ R | a · b ∈ I for all b ∈ J} .

Decide whether each of following statements is true or false. Give reasons for your answers. (a) (b) (c) (d) Every finitely generated module over an Artinian ring is Artinian. Every Artinian module is finitely generated. Every ring has a module that is both Noetherian and Artinian. , we have an additive and a multiplicative commutative monoid, and a distributive law). 10 (The ring of analytic functions). , all functions that are given by power series that converge on all of R. Show that R is not Noetherian.

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