By Gregor Kemper

This textbook bargains a radical, sleek advent into commutative algebra. it's intented frequently to function a advisor for a process one or semesters, or for self-study. The conscientiously chosen subject material concentrates at the options and effects on the middle of the sphere. The publication continues a continuing view at the typical geometric context, allowing the reader to achieve a deeper knowing of the cloth. even though it emphasizes conception, 3 chapters are dedicated to computational facets. Many illustrative examples and workouts improve the text.

**Read Online or Download A Course in Commutative Algebra PDF**

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

**Example text**

So indeed I = I is a ﬁnitely generated ideal. 4. 4, we obtain the following corollary. 12 (Finitely generated algebras). Every ﬁnitely generated algebra over a Noetherian ring is Noetherian. In particular, every aﬃne algebra is Noetherian. A special case is the celebrated basis theorem of Hilbert. 13 (Hilbert’s basis theorem). Let K be a ﬁeld. Then the polynomial ring K[x1 , . . , xn ] is Noetherian. In particular, every ideal in K[x1 , . . , xn ] is ﬁnitely generated. The name basis theorem comes from the fact that generating sets of ideals are sometimes called bases.

6, and since ∅ = VSpec(R) ({1}) and Spec(R) = VSpec(R) (∅), this indeed deﬁnes a topology. A subset of Spec(R) is equipped with the subspace topology induced from the Zariski topology on Spec(R). The following proposition contains all the important general facts about the maps VSpec(R) and IR deﬁned above. 19. 6 (Properties of VSpec(R) and IR ). Let R be a ring. (a) Let S, T ⊆ R be subsets. Then VSpec(R) (S) ∪ VSpec(R) (T ) = VSpec(R) (S)R ∩ (T )R . 2 Spectra 37 (b) Let M be a nonempty set of subsets of R.

Xn ] with Rd the space of all homogeneous polynomials of degree d (including the zero polynomial). Let R be graded and set Rd , I= d∈N>0 which obviously is an ideal. Sometimes I is called the irrelevant ideal. Prove the equivalence of the following statements. (a) R is Noetherian. (b) R0 is Noetherian and I is ﬁnitely generated. (c) R0 is Noetherian and R is ﬁnitely generated as an R0 -algebra. 12, a ﬁnitely generated algebra over a Noetherian rings is Noetherian. However, Noetherian algebras are not always ﬁnitely generated.