By Gregor Kemper

ISBN-10: 3642035442

ISBN-13: 9783642035449

ISBN-10: 3642035450

ISBN-13: 9783642035456

This textbook bargains an intensive, smooth advent into commutative algebra. it truly is intented as a rule to function a consultant for a process one or semesters, or for self-study. The rigorously chosen material concentrates at the innovations and effects on the heart of the sector. The booklet keeps a relentless view at the typical geometric context, permitting the reader to achieve a deeper knowing of the cloth. even though it emphasizes concept, 3 chapters are dedicated to computational facets. Many illustrative examples and workouts increase the text.

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**Extra resources for A Course in Commutative Algebra**

**Example text**

To get the proof for the case of Artinian modules, replace every occurrence of the word “ascending” in the above argument by “descending,” and exchange “Mi ” and “Mn ” in the proof of Mi = Mn . We need the following deﬁnition to push the theory further. 5 (Ideal product). Let R be a ring, I ⊆ R and ideal, and M an R-module. (a) The product of I and M is deﬁned to be the abelian group generated by all products a · m of elements from I and elements from M . So n ai mi n ∈ N, ai ∈ I, and mi ∈ M . IM = i=1 Clearly IM ⊆ M is a submodule.

Therefore there exists y ∈ J with xy = x. By the deﬁnition of J, y lies in every maximal ideal of R, and so y − 1 lies in no maximal ideal. This means that y − 1 is invertible, and (y − 1)x = 0 implies x = 0. This contradicts xJ = {0}. We conclude that J = {0}. 7 and get that R is Noetherian and that every prime ideal is maximal. 8 raises the question whether it is also true that every Artinian module over a ring is Noetherian. 2. 2 Noetherian Rings and Modules The following theorem gives an alternative deﬁnition of Noetherian modules.

To the irreducible components of X (if X is Noetherian). 14 (Minimal prime ideals). Let R be a Noetherian ring. (a) There exist only ﬁnitely many minimal prime ideals P1 , . . , Pn of R. (b) Every prime ideal of R contains at least one of the Pi . (c) The nilradical is the intersection of the Pi : n nil(R) = Pi . i=1 (d) Let I ⊆ R be an ideal. Then the set VSpec(R) (I) has ﬁnitely many minimal elements Q1 , . . , Qr , and √ I= r Qi . i=1 Proof. 10(b), the (maximal) closed, irreducible subsets of X := Spec(R) correspond to (minimal) prime ideals 42 3 The Zariski Topology of R.

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